### The Mizar article:

### Kuratowski - Zorn Lemma

**by****Wojciech A. Trybulec, and****Grzegorz Bancerek**

- Received September 19, 1989
Copyright (c) 1989 Association of Mizar Users

- MML identifier: ORDERS_2
- [ MML identifier index ]

environ vocabulary RELAT_1, FUNCT_1, ORDERS_1, ORDINAL1, BOOLE, RELAT_2, WELLORD1, TARSKI, ZFMISC_1, WELLORD2, SETFAM_1, ORDERS_2, HAHNBAN, PARTFUN1; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, FUNCT_1, RELAT_2, RELSET_1, PARTFUN1, WELLORD1, SETFAM_1, STRUCT_0, ORDERS_1, ORDINAL1, WELLORD2; constructors RELAT_2, WELLORD1, ORDERS_1, WELLORD2, PARTFUN1, XBOOLE_0; clusters ORDERS_1, ORDINAL1, RELSET_1, STRUCT_0, SUBSET_1, ZFMISC_1, XBOOLE_0; requirements BOOLE, SUBSET; definitions RELAT_2, TARSKI, WELLORD1, RELAT_1, ORDERS_1, ORDINAL1, XBOOLE_0; theorems ORDERS_1, RELAT_1, RELAT_2, RELSET_1, TARSKI, WELLORD1, ZFMISC_1, FUNCT_1, ORDINAL1, WELLORD2, SETFAM_1, XBOOLE_0, XBOOLE_1, PARTFUN1; schemes RELSET_1, TARSKI, RELAT_1, FUNCT_1, ORDINAL1, XBOOLE_0; begin reserve R,P for Relation, X,X1,X2,Y,Z,x,y,z,u for set, g,h for Function, O for Order of X, D for non empty set, d,d1,d2 for Element of D, A for non empty Poset, C for Chain of A, S for Subset of A, a,a1,a2,a3,a4,b,c1,c2 for Element of A, A1,A2,B for Ordinal, L,L1,L2 for T-Sequence; :: :: Orders. :: theorem Th1: dom O = X & rng O = X proof thus dom O = X proof thus dom O c= X; let x; assume x in X; then [x,x] in O by ORDERS_1:12; hence thesis by RELAT_1:def 4; end; thus rng O c= X; let x; assume x in X; then [x,x] in O by ORDERS_1:12; hence thesis by RELAT_1:def 5; end; theorem Th2: field O = X proof thus X = X \/ X .= dom O \/ X by Th1 .= dom O \/ rng O by Th1 .= field O by RELAT_1:def 6; end; definition let R; attr R is being_quasi-order means R is reflexive transitive; synonym R is_quasi-order; attr R is being_partial-order means :Def2: R is reflexive transitive antisymmetric; synonym R is_partial-order; attr R is being_linear-order means :Def3: R is reflexive transitive antisymmetric connected; synonym R is_linear-order; end; canceled 3; theorem R is_quasi-order implies R~ is_quasi-order proof assume R is reflexive & R is transitive; hence R~ is reflexive & R~ is transitive by RELAT_2:27,42; end; theorem R is_partial-order implies R~ is_partial-order proof assume R is reflexive & R is transitive & R is antisymmetric; hence R~ is reflexive & R~ is transitive & R~ is antisymmetric by RELAT_2:27,40,42; end; Lm1: R is connected implies R~ is connected proof assume A1: x in field R & y in field R & x <> y implies [x,y] in R or [y,x] in R; A2: field R = field(R~) by RELAT_1:38; let x,y; assume x in field(R~) & y in field(R~) & x <> y; then [x,y] in R or [y,x] in R by A1,A2; hence thesis by RELAT_1:def 7; end; theorem Th8: R is_linear-order implies R~ is_linear-order proof assume R is reflexive & R is transitive & R is antisymmetric & R is connected; hence R~ is reflexive & R~ is transitive & R~ is antisymmetric & R~ is connected by Lm1,RELAT_2:27,40,42; end; theorem R is well-ordering implies R is_quasi-order & R is_partial-order & R is_linear-order proof assume R is reflexive & R is transitive & R is antisymmetric & R is connected & R is well_founded; hence R is reflexive & R is transitive & R is reflexive & R is transitive & R is antisymmetric & R is reflexive & R is transitive & R is antisymmetric & R is connected; end; theorem R is_linear-order implies R is_quasi-order & R is_partial-order proof assume R is reflexive & R is transitive & R is antisymmetric & R is connected; hence R is reflexive & R is transitive & R is reflexive & R is transitive & R is antisymmetric; end; theorem Th11: R is_partial-order implies R is_quasi-order proof assume R is reflexive & R is transitive & R is antisymmetric; hence R is reflexive & R is transitive; end; theorem Th12: O is_partial-order by Def2; theorem O is_quasi-order proof O is_partial-order by Th12; hence thesis by Th11; end; theorem O is connected implies O is_linear-order by Def3; Lm2: R c= [:field R,field R:] proof let x; assume A1: x in R; then consider y,z such that A2: x = [y,z] by RELAT_1:def 1; y in field R & z in field R by A1,A2,RELAT_1:30; hence thesis by A2,ZFMISC_1:106; end; Lm3: R is reflexive & X c= field R implies field(R|_2 X) = X proof assume that A1: x in field R implies [x,x] in R and A2: X c= field R; thus field(R|_2 X) c= X proof let y; thus thesis by WELLORD1:19; end; let y; assume y in X; then [y,y] in R & [y,y] in [:X,X:] by A1,A2,ZFMISC_1:106; then [y,y] in R|_2 X by WELLORD1:16; hence thesis by RELAT_1:30; end; theorem R is_quasi-order implies R|_2 X is_quasi-order proof assume R is reflexive & R is transitive; hence R|_2 X is reflexive & R|_2 X is transitive by WELLORD1:22,24; end; theorem Th16: R is_partial-order implies R|_2 X is_partial-order proof assume R is reflexive & R is transitive & R is antisymmetric; hence R|_2 X is reflexive & R|_2 X is transitive & R|_2 X is antisymmetric by WELLORD1:22,24,25; end; theorem R is_linear-order implies R|_2 X is_linear-order proof assume R is reflexive & R is transitive & R is antisymmetric & R is connected; hence R|_2 X is reflexive & R|_2 X is transitive & R|_2 X is antisymmetric & R|_2 X is connected by WELLORD1:22,23,24,25; end; theorem Th18: field((the InternalRel of A) |_2 S) = S proof set P = (the InternalRel of A) |_2 S; thus field P c= S by WELLORD1:20; thus S c= field P proof let x; assume A1: x in S; S c= the carrier of A & the InternalRel of A is_reflexive_in the carrier of A by ORDERS_1:def 4; then [x,x] in the InternalRel of A & [x,x] in [:S,S:] by A1,RELAT_2:def 1,ZFMISC_1:106; then [x,x] in P by WELLORD1:16; then x in dom P & x in rng P by RELAT_1:def 4,def 5; then x in dom P \/ rng P by XBOOLE_0:def 2; hence thesis by RELAT_1:def 6; end; end; theorem (the InternalRel of A) |_2 S is_linear-order implies S is Chain of A proof assume (the InternalRel of A) |_2 S is_linear-order; then (the InternalRel of A) |_2 S is connected & field((the InternalRel of A) |_2 S) = S by Def3,Th18; then A1: (the InternalRel of A) |_2 S is_connected_in S by RELAT_2:def 14; S is strongly_connected proof let x,y; assume A2: x in S & y in S; then reconsider a = x, b = y as Element of A; now per cases; suppose x = y; then a <= b; hence thesis by ORDERS_1:def 9; suppose x <> y; then [x,y] in (the InternalRel of A) |_2 S or [y,x] in (the InternalRel of A) |_2 S by A1,A2,RELAT_2:def 6; hence thesis by WELLORD1:16; end; hence thesis; end; hence thesis; end; theorem (the InternalRel of A) |_2 C is_linear-order proof set P = (the InternalRel of A) |_2 C; the InternalRel of A is_partial-order by Th12; then P is_partial-order by Th16; hence P is reflexive & P is transitive & P is antisymmetric by Def2; A1: C = field P by Th18; P is_connected_in C proof let x,y; assume that A2: x in C & y in C; the InternalRel of A is_strongly_connected_in C by ORDERS_1:def 11; then A3: [x,y] in the InternalRel of A or [y,x] in the InternalRel of A by A2,RELAT_2: def 7; [x,y] in [:C,C:] & [y,x] in [:C,C:] by A2,ZFMISC_1:106; hence thesis by A3,WELLORD1:16; end; hence thesis by A1,RELAT_2:def 14; end; theorem Th21: {} is_quasi-order & {} is_partial-order & {} is_linear-order & {} is well-ordering proof A1: field {} = {} \/ {} by RELAT_1:60,def 6 .= {}; thus A2: {} is reflexive proof let x; assume x in field {}; hence thesis by A1; end; thus A3: {} is transitive proof let x,y,z such that x in field {}; thus thesis; end; hence {} is reflexive & {} is transitive by A2; thus A4: {} is antisymmetric proof let x,y such that x in field {}; thus thesis; end; hence {} is reflexive & {} is transitive & {} is antisymmetric by A2,A3; thus {} is connected proof let x,y such that x in field {}; thus thesis by A1; end; hence {} is reflexive & {} is transitive & {} is antisymmetric & {} is connected by A2,A3,A4; let Y; assume Y c= field {} & Y <> {}; hence thesis by A1,XBOOLE_1:3; end; theorem Th22: id X is_quasi-order & id X is_partial-order proof thus id X is reflexive & id X is transitive & id X is reflexive & id X is transitive & id X is antisymmetric by RELAT_2:23,24; end; definition let R,X; pred R quasi_orders X means :Def4: R is_reflexive_in X & R is_transitive_in X; pred R partially_orders X means :Def5: R is_reflexive_in X & R is_transitive_in X & R is_antisymmetric_in X; pred R linearly_orders X means :Def6: R is_reflexive_in X & R is_transitive_in X & R is_antisymmetric_in X & R is_connected_in X; end; canceled 3; theorem Th26: R well_orders X implies R quasi_orders X & R partially_orders X & R linearly_orders X proof assume R is_reflexive_in X & R is_transitive_in X & R is_antisymmetric_in X & R is_connected_in X & R is_well_founded_in X; hence R is_reflexive_in X & R is_transitive_in X & R is_reflexive_in X & R is_transitive_in X & R is_antisymmetric_in X & R is_reflexive_in X & R is_transitive_in X & R is_antisymmetric_in X & R is_connected_in X; end; theorem Th27: R linearly_orders X implies R quasi_orders X & R partially_orders X proof assume R is_reflexive_in X & R is_transitive_in X & R is_antisymmetric_in X & R is_connected_in X; hence R is_reflexive_in X & R is_transitive_in X & R is_reflexive_in X & R is_transitive_in X & R is_antisymmetric_in X; end; theorem R partially_orders X implies R quasi_orders X proof assume R is_reflexive_in X & R is_transitive_in X & R is_antisymmetric_in X; hence R is_reflexive_in X & R is_transitive_in X; end; theorem Th29: R is_quasi-order implies R quasi_orders field R proof assume R is_reflexive_in field R & R is_transitive_in field R; hence R is_reflexive_in field R & R is_transitive_in field R; end; theorem R quasi_orders Y & X c= Y implies R quasi_orders X proof assume R is_reflexive_in Y & R is_transitive_in Y & X c= Y; hence R is_reflexive_in X & R is_transitive_in X by ORDERS_1:93,95; end; Lm4: R is_reflexive_in X implies R|_2 X is reflexive proof assume A1: x in X implies [x,x] in R; let x; assume x in field(R|_2 X); then x in X by WELLORD1:19; then [x,x] in R & [x,x] in [:X,X:] by A1,ZFMISC_1:106; hence thesis by WELLORD1:16; end; Lm5: R is_transitive_in X implies R|_2 X is transitive proof assume A1: x in X & y in X & z in X & [x,y] in R & [y,z] in R implies [x,z] in R; let x,y,z; assume x in field(R|_2 X) & y in field(R|_2 X) & z in field(R|_2 X); then A2: x in X & y in X & z in X by WELLORD1:19; assume [x,y] in R|_2 X & [y,z] in R|_2 X; then [x,y] in R & [y,z] in R by WELLORD1:16; then [x,z] in R & [x,z] in [:X,X:] by A1,A2,ZFMISC_1:106; hence thesis by WELLORD1:16; end; Lm6: R is_antisymmetric_in X implies R|_2 X is antisymmetric proof assume A1: x in X & y in X & [x,y] in R & [y,x] in R implies x = y; let x,y; assume x in field(R|_2 X) & y in field(R|_2 X); then A2: x in X & y in X by WELLORD1:19; assume [x,y] in R|_2 X & [y,x] in R|_2 X; then [x,y] in R & [y,x] in R by WELLORD1:16; hence thesis by A1,A2; end; Lm7: R is_connected_in X implies R|_2 X is connected proof assume A1: x in X & y in X & x <> y implies [x,y] in R or [y,x] in R; let x,y; assume x in field(R|_2 X) & y in field(R|_2 X) & x <> y; then x in X & y in X & x <> y by WELLORD1:19; then ([x,y] in R or [y,x] in R) & [x,y] in [:X,X:] & [y,x] in [:X,X:] by A1,ZFMISC_1:106; hence thesis by WELLORD1:16; end; theorem R quasi_orders X implies R|_2 X is_quasi-order proof assume R is_reflexive_in X & R is_transitive_in X; hence R|_2 X is reflexive & R|_2 X is transitive by Lm4,Lm5; end; theorem Th32: R is_partial-order implies R partially_orders field R proof assume R is_reflexive_in field R & R is_transitive_in field R & R is_antisymmetric_in field R; hence R is_reflexive_in field R & R is_transitive_in field R & R is_antisymmetric_in field R; end; theorem R partially_orders Y & X c= Y implies R partially_orders X proof assume R is_reflexive_in Y & R is_transitive_in Y & R is_antisymmetric_in Y & X c= Y; hence R is_reflexive_in X & R is_transitive_in X & R is_antisymmetric_in X by ORDERS_1:93,94,95; end; theorem R partially_orders X implies R|_2 X is_partial-order proof assume R is_reflexive_in X & R is_transitive_in X & R is_antisymmetric_in X; hence R|_2 X is reflexive & R|_2 X is transitive & R|_2 X is antisymmetric by Lm4,Lm5,Lm6; end; Lm8: R is_connected_in X & Y c= X implies R is_connected_in Y proof assume that A1: R is_connected_in X and A2: Y c= X; let x,y; assume x in Y & y in Y; hence thesis by A1,A2,RELAT_2:def 6; end; theorem R is_linear-order implies R linearly_orders field R proof assume R is_reflexive_in field R & R is_transitive_in field R & R is_antisymmetric_in field R & R is_connected_in field R; hence R is_reflexive_in field R & R is_transitive_in field R & R is_antisymmetric_in field R & R is_connected_in field R; end; theorem R linearly_orders Y & X c= Y implies R linearly_orders X proof assume R is_reflexive_in Y & R is_transitive_in Y & R is_antisymmetric_in Y & R is_connected_in Y & X c= Y; hence R is_reflexive_in X & R is_transitive_in X & R is_antisymmetric_in X & R is_connected_in X by Lm8,ORDERS_1:93,94,95; end; theorem R linearly_orders X implies R|_2 X is_linear-order proof assume R is_reflexive_in X & R is_transitive_in X & R is_antisymmetric_in X & R is_connected_in X; hence R|_2 X is reflexive & R|_2 X is transitive & R|_2 X is antisymmetric & R|_2 X is connected by Lm4,Lm5,Lm6,Lm7; end; Lm9: R is_reflexive_in X implies R~ is_reflexive_in X proof assume A1: x in X implies [x,x] in R; let x; assume x in X; then [x,x] in R by A1; hence thesis by RELAT_1:def 7; end; Lm10: R is_transitive_in X implies R~ is_transitive_in X proof assume A1: x in X & y in X & z in X & [x,y] in R & [y,z] in R implies [x,z] in R; let x,y,z; assume A2: x in X & y in X & z in X & [x,y] in R~ & [y,z] in R~; then [y,x] in R & [z,y] in R by RELAT_1:def 7; then [z,x] in R by A1,A2; hence thesis by RELAT_1:def 7; end; Lm11: R is_antisymmetric_in X implies R~ is_antisymmetric_in X proof assume A1: x in X & y in X & [x,y] in R & [y,x] in R implies x = y; let x,y; assume A2: x in X & y in X & [x,y] in R~ & [y,x] in R~; then [x,y] in R & [y,x] in R by RELAT_1:def 7; hence thesis by A1,A2; end; Lm12: R is_connected_in X implies R~ is_connected_in X proof assume A1: x in X & y in X & x <> y implies [x,y] in R or [y,x] in R; let x,y; assume x in X & y in X & x <> y; then [x,y] in R or [y,x] in R by A1; hence thesis by RELAT_1:def 7; end; theorem R quasi_orders X implies R~ quasi_orders X proof assume R is_reflexive_in X & R is_transitive_in X; hence R~ is_reflexive_in X & R~ is_transitive_in X by Lm9,Lm10; end; theorem Th39: R partially_orders X implies R~ partially_orders X proof assume R is_reflexive_in X & R is_transitive_in X & R is_antisymmetric_in X; hence R~ is_reflexive_in X & R~ is_transitive_in X & R~ is_antisymmetric_in X by Lm9,Lm10,Lm11; end; theorem R linearly_orders X implies R~ linearly_orders X proof assume R is_reflexive_in X & R is_transitive_in X & R is_antisymmetric_in X & R is_connected_in X; hence R~ is_reflexive_in X & R~ is_transitive_in X & R~ is_antisymmetric_in X & R~ is_connected_in X by Lm9,Lm10,Lm11,Lm12; end; theorem O quasi_orders X proof field O = X by ORDERS_1:97; then O is_reflexive_in X & O is_transitive_in X by RELAT_2:def 9,def 16; hence thesis by Def4; end; theorem O partially_orders X proof field O = X by ORDERS_1:97; then O is_reflexive_in X & O is_antisymmetric_in X & O is_transitive_in X by RELAT_2:def 9,def 12,def 16; hence thesis by Def5; end; theorem Th43: R partially_orders X implies R |_2 X is Order of X proof assume A1: R partially_orders X; then A2: R is_reflexive_in X by Def5; set S = R |_2 X; A3: R |_2 X is_reflexive_in X proof let x; assume x in X; then [x,x] in [:X,X:] & [x,x] in R by A2,RELAT_2:def 1,ZFMISC_1:106; hence [x,x] in R |_2 X by WELLORD1:16; end; A4: field S c= X by WELLORD1:20; field S = dom S \/ rng S by RELAT_1:def 6; then dom S c= field S & rng S c= field S by XBOOLE_1:7; then dom S c= X & rng S c= X by A4,XBOOLE_1:1; then reconsider S as Relation of X by RELSET_1:11; A5: dom S = X by A3,ORDERS_1:98; A6: field S = X by A3,ORDERS_1:98; A7: R |_2 X is_antisymmetric_in X proof let x,y; assume that A8: x in X & y in X and A9: [x,y] in R |_2 X & [y,x] in R |_2 X; [x,y] in R & [y,x] in R & R is_antisymmetric_in X by A1,A9,Def5,WELLORD1:16; hence thesis by A8,RELAT_2:def 4; end; A10: R |_2 X is_transitive_in X proof let x,y,z; assume that A11: x in X & y in X & z in X and A12: [x,y] in R |_2 X & [y,z] in R |_2 X; [x,y] in R & [y,z] in R & R is_transitive_in X by A1,A12,Def5,WELLORD1:16; then [x,z] in R & [x,z] in [:X,X:] by A11,RELAT_2:def 8,ZFMISC_1:106; hence thesis by WELLORD1:16; end; A13: S is total by A5,PARTFUN1:def 4; A14: S is reflexive by A3,A6,RELAT_2:def 9; A15: S is antisymmetric by A6,A7,RELAT_2:def 12; S is transitive by A6,A10,RELAT_2:def 16; hence thesis by A13,A14,A15; end; theorem R linearly_orders X implies R |_2 X is Order of X proof assume R linearly_orders X; then R partially_orders X by Th27; hence thesis by Th43; end; theorem R well_orders X implies R |_2 X is Order of X proof assume R well_orders X; then R partially_orders X by Th26; hence thesis by Th43; end; theorem the InternalRel of A linearly_orders S implies S is Chain of A proof assume the InternalRel of A linearly_orders S; then the InternalRel of A is_reflexive_in S & the InternalRel of A is_connected_in S by Def6; then the InternalRel of A is_strongly_connected_in S by ORDERS_1:92; hence thesis by ORDERS_1:def 11; end; theorem the InternalRel of A linearly_orders C proof the InternalRel of A is_reflexive_in the carrier of A & the InternalRel of A is_transitive_in the carrier of A & the InternalRel of A is_antisymmetric_in the carrier of A & C c= the carrier of A by ORDERS_1:def 4,def 5,def 6; hence the InternalRel of A is_reflexive_in C & the InternalRel of A is_transitive_in C & the InternalRel of A is_antisymmetric_in C by ORDERS_1:93,94,95; the InternalRel of A is_strongly_connected_in C by ORDERS_1:def 11; hence thesis by ORDERS_1:92; end; theorem id X quasi_orders X & id X partially_orders X proof A1: field id X = dom id X \/ rng id X by RELAT_1:def 6 .= X \/ rng id X by RELAT_1:71 .= X \/ X by RELAT_1:71 .= X; id X is_quasi-order & id X is_partial-order by Th22; hence thesis by A1,Th29,Th32; end; definition let R,X; pred X has_upper_Zorn_property_wrt R means :Def7: for Y st Y c= X & R|_2 Y is_linear-order ex x st x in X & for y st y in Y holds [y,x] in R; pred X has_lower_Zorn_property_wrt R means for Y st Y c= X & R|_2 Y is_linear-order ex x st x in X & for y st y in Y holds [x,y] in R; end; Lm13: (R|_2 X)~ = R~|_2 X proof now let x,y; thus [x,y] in R~|_2 X implies [y,x] in R|_2 X proof assume [x,y] in R~|_2 X; then [x,y] in R~ & [x,y] in [:X,X:] by WELLORD1:16; then [y,x] in R & [y,x] in [:X,X:] by RELAT_1:def 7,ZFMISC_1:107; hence thesis by WELLORD1:16; end; assume [y,x] in R|_2 X; then [y,x] in R & [y,x] in [:X,X:] by WELLORD1:16; then [x,y] in R~ & [x,y] in [:X,X:] by RELAT_1:def 7,ZFMISC_1:107; hence [x,y] in R~|_2 X by WELLORD1:16; end; hence thesis by RELAT_1:def 7; end; Lm14: now let R; thus R|_2 {} = {}|R|({} qua set) by WELLORD1:17 .= {} by RELAT_1:110; end; canceled 2; theorem Th51: X has_upper_Zorn_property_wrt R implies X <> {} proof assume A1: for Y st Y c= X & R|_2 Y is_linear-order ex x st x in X & for y st y in Y holds [y,x] in R; {} c= X & R|_2 {} is_linear-order by Lm14,Th21,XBOOLE_1:2; then ex x st x in X & for y st y in {} holds [y,x] in R by A1; hence thesis; end; theorem X has_lower_Zorn_property_wrt R implies X <> {} proof assume A1: for Y st Y c= X & R|_2 Y is_linear-order ex x st x in X & for y st y in Y holds [x,y] in R; {} c= X & R|_2 {} is_linear-order by Lm14,Th21,XBOOLE_1:2; then ex x st x in X & for y st y in {} holds [x,y] in R by A1; hence thesis; end; theorem Th53: X has_upper_Zorn_property_wrt R iff X has_lower_Zorn_property_wrt R~ proof thus X has_upper_Zorn_property_wrt R implies X has_lower_Zorn_property_wrt R~ proof assume A1: for Y st Y c= X & R|_2 Y is_linear-order ex x st x in X & for y st y in Y holds [y,x] in R; let Y; A2: R~|_2 Y = (R|_2 Y)~ & (R|_2 Y)~~ = R|_2 Y by Lm13; assume A3: Y c= X & R~|_2 Y is_linear-order; then R|_2 Y is_linear-order by A2,Th8; then consider x such that A4: x in X & for y st y in Y holds [y,x] in R by A1,A3; take x; thus x in X by A4; let y; assume y in Y; then [y,x] in R by A4; hence [x,y] in R~ by RELAT_1:def 7; end; assume A5: for Y st Y c= X & R~|_2 Y is_linear-order ex x st x in X & for y st y in Y holds [x,y] in R~; let Y; A6: R~|_2 Y = (R|_2 Y)~ & (R|_2 Y)~~ = R|_2 Y by Lm13; assume A7: Y c= X & R|_2 Y is_linear-order; then R~|_2 Y is_linear-order by A6,Th8; then consider x such that A8: x in X & for y st y in Y holds [x,y] in R~ by A5,A7; take x; thus x in X by A8; let y; assume y in Y; then [x,y] in R~ by A8; hence [y,x] in R by RELAT_1:def 7; end; theorem X has_upper_Zorn_property_wrt R~ iff X has_lower_Zorn_property_wrt R proof R~~ = R; hence thesis by Th53; end; definition let R,x; pred x is_maximal_in R means :Def9: x in field R & not ex y st y in field R & y <> x & [x,y] in R; pred x is_minimal_in R means :Def10: x in field R & not ex y st y in field R & y <> x & [y,x] in R; pred x is_superior_of R means :Def11: x in field R & for y st y in field R & y <> x holds [y,x] in R; pred x is_inferior_of R means :Def12: x in field R & for y st y in field R & y <> x holds [x,y] in R; end; canceled 4; theorem x is_inferior_of R & R is antisymmetric implies x is_minimal_in R proof assume that A1: x is_inferior_of R and A2: R is antisymmetric; thus A3: x in field R by A1,Def12; let y; assume that A4: y in field R and A5: y <> x and A6: [y,x] in R; [x,y] in R & R is_antisymmetric_in field R by A1,A2,A4,A5,Def12,RELAT_2: def 12; hence thesis by A3,A4,A5,A6,RELAT_2:def 4; end; theorem x is_superior_of R & R is antisymmetric implies x is_maximal_in R proof assume that A1: x is_superior_of R and A2: R is antisymmetric; thus A3: x in field R by A1,Def11; let y; assume that A4: y in field R and A5: y <> x and A6: [x,y] in R; [y,x] in R & R is_antisymmetric_in field R by A1,A2,A4,A5,Def11,RELAT_2: def 12; hence thesis by A3,A4,A5,A6,RELAT_2:def 4; end; theorem x is_minimal_in R & R is connected implies x is_inferior_of R proof assume that A1: x is_minimal_in R and A2: R is connected; thus A3: x in field R by A1,Def10; A4: R is_connected_in field R by A2,RELAT_2:def 14; let y; assume that A5: y in field R and A6: y <> x; [x,y] in R or [y,x] in R by A3,A4,A5,A6,RELAT_2:def 6; hence thesis by A1,A5,A6,Def10; end; theorem x is_maximal_in R & R is connected implies x is_superior_of R proof assume that A1: x is_maximal_in R and A2: R is connected; thus A3: x in field R by A1,Def9; A4: R is_connected_in field R by A2,RELAT_2:def 14; let y; assume that A5: y in field R and A6: y <> x; [x,y] in R or [y,x] in R by A3,A4,A5,A6,RELAT_2:def 6; hence thesis by A1,A5,A6,Def9; end; theorem x in X & x is_superior_of R & X c= field R & R is reflexive implies X has_upper_Zorn_property_wrt R proof assume A1: x in X & x is_superior_of R & X c= field R & R is_reflexive_in field R; let Y such that A2: Y c= X & R|_2 Y is_linear-order; take x; thus x in X by A1; let y; assume y in Y; then y in X by A2; then y in field R & (y = x or y <> x) by A1; hence [y,x] in R by A1,Def11,RELAT_2:def 1; end; theorem x in X & x is_inferior_of R & X c= field R & R is reflexive implies X has_lower_Zorn_property_wrt R proof assume A1: x in X & x is_inferior_of R & X c= field R & R is_reflexive_in field R; let Y such that A2: Y c= X & R|_2 Y is_linear-order; take x; thus x in X by A1; let y; assume y in Y; then y in X by A2; then y in field R & (y = x or y <> x) by A1; hence [x,y] in R by A1,Def12,RELAT_2:def 1; end; theorem Th65: x is_minimal_in R iff x is_maximal_in R~ proof A1: field R = field(R~) by RELAT_1:38; thus x is_minimal_in R implies x is_maximal_in R~ proof assume A2: x in field R & not ex y st y in field R & y <> x & [y,x] in R; hence x in field(R~) by RELAT_1:38; let y; assume y in field(R~) & y <> x; then not [y,x] in R by A1,A2; hence not [x,y] in R~ by RELAT_1:def 7; end; assume A3: x in field(R~) & not ex y st y in field(R~) & y <> x & [x,y] in R~; hence x in field R by RELAT_1:38; let y; assume y in field R & y <> x; then not [x,y] in R~ by A1,A3; hence not [y,x] in R by RELAT_1:def 7; end; theorem x is_minimal_in R~ iff x is_maximal_in R proof A1: field R = field(R~) by RELAT_1:38; thus x is_minimal_in R~ implies x is_maximal_in R proof assume A2: x in field(R~) & not ex y st y in field(R~) & y <> x & [y,x] in R~; hence x in field R by RELAT_1:38; let y; assume y in field R & y <> x; then not [y,x] in R~ by A1,A2; hence not [x,y] in R by RELAT_1:def 7; end; assume A3: x in field R & not ex y st y in field R & y <> x & [x,y] in R; hence x in field(R~) by RELAT_1:38; let y; assume y in field(R~) & y <> x; then not [x,y] in R by A1,A3; hence not [y,x] in R~ by RELAT_1:def 7; end; theorem x is_inferior_of R iff x is_superior_of R~ proof A1: field R = field(R~) by RELAT_1:38; thus x is_inferior_of R implies x is_superior_of R~ proof assume A2: x in field R & for y st y in field R & y <> x holds [x,y] in R; hence x in field(R~) by RELAT_1:38; let y; assume y in field(R~) & y <> x; then [x,y] in R by A1,A2; hence [y,x] in R~ by RELAT_1:def 7; end; assume A3: x in field(R~) & for y st y in field(R~) & y <> x holds [y,x] in R~; hence x in field R by RELAT_1:38; let y; assume y in field R & y <> x; then [y,x] in R~ by A1,A3; hence [x,y] in R by RELAT_1:def 7; end; theorem x is_inferior_of R~ iff x is_superior_of R proof A1: field R = field(R~) by RELAT_1:38; thus x is_inferior_of R~ implies x is_superior_of R proof assume A2: x in field(R~) & for y st y in field(R~) & y <> x holds [x,y] in R~; hence x in field R by RELAT_1:38; let y; assume y in field R & y <> x; then [x,y] in R~ by A1,A2; hence [y,x] in R by RELAT_1:def 7; end; assume A3: x in field R & for y st y in field R & y <> x holds [y,x] in R; hence x in field(R~) by RELAT_1:38; let y; assume y in field(R~) & y <> x; then [y,x] in R by A1,A3; hence [x,y] in R~ by RELAT_1:def 7; end; theorem a is_minimal_in the InternalRel of A iff for b holds not b < a proof A1: the carrier of A = field(the InternalRel of A) by Th2; thus a is_minimal_in the InternalRel of A implies for b holds not b < a proof assume A2: a is_minimal_in the InternalRel of A; let b; a = b or not [b,a] in the InternalRel of A by A1,A2,Def10; then a = b or not b <= a by ORDERS_1:def 9; hence thesis by ORDERS_1:def 10; end; assume A3: for b holds not b < a; thus a in field(the InternalRel of A) by A1; let y; assume that A4: y in field(the InternalRel of A) and A5: y <> a and A6: [y,a] in the InternalRel of A; y is Element of A by A4,Th2; then reconsider b = y as Element of A; b <= a by A6,ORDERS_1:def 9; then b < a by A5,ORDERS_1:def 10; hence thesis by A3; end; theorem a is_maximal_in the InternalRel of A iff for b holds not a < b proof A1: the carrier of A = field(the InternalRel of A) by Th2; thus a is_maximal_in the InternalRel of A implies for b holds not a < b proof assume A2: a is_maximal_in the InternalRel of A; let b; a = b or not [a,b] in the InternalRel of A by A1,A2,Def9; then a = b or not a <= b by ORDERS_1:def 9; hence thesis by ORDERS_1:def 10; end; assume A3: for b holds not a < b; thus a in field(the InternalRel of A) by A1; let y; assume that A4: y in field(the InternalRel of A) and A5: y <> a and A6: [a,y] in the InternalRel of A; y is Element of A by A4,Th2; then reconsider b = y as Element of A; a <= b by A6,ORDERS_1:def 9; then a < b by A5,ORDERS_1:def 10; hence thesis by A3; end; theorem a is_superior_of the InternalRel of A iff for b st a <> b holds b < a proof A1: the carrier of A = field(the InternalRel of A) by Th2; thus a is_superior_of the InternalRel of A implies for b st a <> b holds b < a proof assume A2: a is_superior_of the InternalRel of A; let b; assume A3: a <> b; then [b,a] in the InternalRel of A by A1,A2,Def11; then b <= a by ORDERS_1:def 9; hence thesis by A3,ORDERS_1:def 10; end; assume A4: for b st a <> b holds b < a; thus a in field(the InternalRel of A) by A1; let y; assume y in field(the InternalRel of A); then y is Element of A by Th2; then reconsider b = y as Element of A; assume y <> a; then b < a by A4; then b <= a by ORDERS_1:def 10; hence thesis by ORDERS_1:def 9; end; theorem a is_inferior_of the InternalRel of A iff for b st a <> b holds a < b proof A1: the carrier of A = field(the InternalRel of A) by Th2; thus a is_inferior_of the InternalRel of A implies for b st a <> b holds a < b proof assume A2: a is_inferior_of the InternalRel of A; let b; assume A3: a <> b; then [a,b] in the InternalRel of A by A1,A2,Def12; then a <= b by ORDERS_1:def 9; hence thesis by A3,ORDERS_1:def 10; end; assume A4: for b st a <> b holds a < b; thus a in field(the InternalRel of A) by A1; let y; assume y in field(the InternalRel of A); then y is Element of A by Th2; then reconsider b = y as Element of A; assume y <> a; then a < b by A4; then a <= b by ORDERS_1:def 10; hence thesis by ORDERS_1:def 9; end; Lm15: R well_orders X & Y c= X implies R well_orders Y proof assume that A1: R well_orders X and A2: Y c= X; R is_reflexive_in X & R is_transitive_in X & R is_antisymmetric_in X & R is_connected_in X by A1,WELLORD1:def 5; hence R is_reflexive_in Y & R is_transitive_in Y & R is_antisymmetric_in Y & R is_connected_in Y by A2,Lm8,ORDERS_1:93,94,95; let Z; assume that A3: Z c= Y and A4: Z <> {}; Z c= X & R is_well_founded_in X by A1,A2,A3,WELLORD1:def 5,XBOOLE_1:1; hence thesis by A4,WELLORD1:def 3; end; :: :: Kuratowski - Zorn Lemma. :: theorem Th73: (for C ex a st for b st b in C holds b <= a) implies ex a st for b holds not a < b proof assume A1: for C ex a st for b st b in C holds b <= a; consider f being Choice_Function of BOOL(the carrier of A); reconsider F = union(Chains(f)) as Chain of f by ORDERS_1:89; consider a such that A2: for b st b in F holds b <= a by A1; take a; let b; assume A3: a < b; now let a1; assume a1 in F; then a1 <= a by A2; hence a1 < b by A3,ORDERS_1:32; end; then b in {a1 : for a2 st a2 in F holds a2 < a1}; then UpperCone(F) <> {} by ORDERS_1:def 12; then not UpperCone(F) in {{}} by TARSKI:def 1; then UpperCone(F) in bool(the carrier of A) \ {{}} by XBOOLE_0:def 4; then UpperCone(F) in BOOL(the carrier of A) & not {} in BOOL(the carrier of A) by ORDERS_1:4,def 2; then A4: f.UpperCone(F) in UpperCone(F) by ORDERS_1:def 1; then reconsider c = f.UpperCone(F) as Element of A; c in {a1 : for a2 st a2 in F holds a2 < a1} by A4,ORDERS_1:def 12; then A5: ex c11 being Element of A st c11 = c & for a2 st a2 in F holds a2 < c11; reconsider Z = F \/ {c} as Subset of A; Z c= the carrier of A & the InternalRel of A is_reflexive_in the carrier of A & the InternalRel of A is_antisymmetric_in the carrier of A & the InternalRel of A is_transitive_in the carrier of A by ORDERS_1:def 4,def 5,def 6; then A6: the InternalRel of A is_reflexive_in Z & the InternalRel of A is_antisymmetric_in Z & the InternalRel of A is_transitive_in Z by ORDERS_1:93,94,95; A7: the InternalRel of A is_connected_in Z proof let x,y; assume that A8: x in Z and A9: y in Z; reconsider x1 = x, y1 = y as Element of A by A8,A9; now per cases by A8,A9,XBOOLE_0:def 2; suppose x1 in F & y1 in F; then x1 <= y1 or y1 <= x1 by ORDERS_1:38; hence thesis by ORDERS_1:def 9; suppose x1 in F & y1 in {c}; then x1 in F & y1 = c by TARSKI:def 1; then x1 < y1 by A5; then x1 <= y1 by ORDERS_1:def 10; hence thesis by ORDERS_1:def 9; suppose x1 in {c} & y1 in F; then y1 in F & x1 = c by TARSKI:def 1; then y1 < x1 by A5; then y1 <= x1 by ORDERS_1:def 10; hence thesis by ORDERS_1:def 9; suppose x1 in {c} & y1 in {c}; then c <= c & x1 = c & y1 = c by TARSKI:def 1; hence thesis; end; hence thesis; end; the InternalRel of A is_well_founded_in Z proof let Y; assume that A10: Y c= Z and A11: Y <> {}; now per cases; case A12: Y = {c}; take x = c; thus x in Y by A12,TARSKI:def 1; assume (the InternalRel of A)-Seg(x) meets Y; then consider x' being set such that A13: x' in (the InternalRel of A)-Seg(x) & x' in Y by XBOOLE_0:3; x' = c & x' in (the InternalRel of A)-Seg(x) by A12,A13,TARSKI:def 1; hence contradiction by WELLORD1:def 1; case A14: Y <> {c}; set X = Y \ {c}; A15: X c= F proof let x; assume that A16: x in X and A17: not x in F; x in Y by A16,XBOOLE_0:def 4; then x in {c} & not x in {c} by A10,A16,A17,XBOOLE_0:def 2, def 4; hence thesis; end; the InternalRel of A well_orders F by ORDERS_1:def 16; then the InternalRel of A well_orders X by A15,Lm15; then A18: the InternalRel of A is_well_founded_in X by WELLORD1:def 5; now assume X = {}; then Y c= {c} by XBOOLE_1:37; hence contradiction by A11,A14,ZFMISC_1:39; end; then consider x such that A19: x in X and A20: (the InternalRel of A)-Seg(x) misses X by A18,WELLORD1:def 3; A21: (the InternalRel of A)-Seg(x) /\ X = {} by A20,XBOOLE_0:def 7; take x' = x; X c= Y by XBOOLE_1:36; hence x' in Y by A19; now per cases; suppose c in Y; then {c} c= Y by ZFMISC_1:37; then A22: Y = X \/ {c} by XBOOLE_1:45; A23: now assume c in (the InternalRel of A)-Seg(x'); then A24: c <> x' & [c,x'] in the InternalRel of A by WELLORD1:def 1; x' in F by A15,A19; then reconsider x'' = x' as Element of A; c <> x'' & c <= x'' by A24,ORDERS_1:def 9; then c < x'' & x'' < c by A5,A15,A19,ORDERS_1:def 10; hence contradiction by ORDERS_1:28; end; now assume A25: (the InternalRel of A)-Seg(x') /\ {c} <> {} ; consider x being Element of (the InternalRel of A)-Seg(x') /\ {c}; x in {c} by A25,XBOOLE_0:def 3; then x = c & x in (the InternalRel of A)-Seg(x') by A25,TARSKI:def 1,XBOOLE_0:def 3; hence contradiction by A23; end; then (the InternalRel of A)-Seg(x') /\ Y = {} \/ {} by A21,A22,XBOOLE_1:23 .= {}; hence (the InternalRel of A)-Seg(x') misses Y by XBOOLE_0:def 7; suppose not c in Y; then Y misses {c} by ZFMISC_1:56; hence (the InternalRel of A)-Seg(x') misses Y by A20,XBOOLE_1:83; end; hence (the InternalRel of A)-Seg(x') misses Y; end; hence thesis; end; then A26: the InternalRel of A well_orders Z by A6,A7,WELLORD1:def 5; the InternalRel of A is_strongly_connected_in Z by A6,A7,ORDERS_1:92; then A27: Z is Chain of A by ORDERS_1:def 11; now let a1; assume A28: a1 in Z; now per cases; suppose A29: a1 = c; InitSegm(Z,c) = F proof thus InitSegm(Z,c) c= F proof let x; assume that A30: x in InitSegm(Z,c) and A31: not x in F; A32: x in LowerCone{c} /\ Z by A30,ORDERS_1:def 14; then x in Z by XBOOLE_0:def 3; then x in {c} by A31,XBOOLE_0:def 2; then x = c & x in LowerCone{c} by A32,TARSKI:def 1,XBOOLE_0: def 3; hence contradiction by ORDERS_1:50; end; let x; assume A33: x in F; then A34: x in Z by XBOOLE_0:def 2; reconsider y = x as Element of A by A33; y < c by A5,A33; then x in LowerCone{c} by ORDERS_1:52; then x in LowerCone{c} /\ Z by A34,XBOOLE_0:def 3; hence thesis by ORDERS_1:def 14; end; hence f.UpperCone(InitSegm(Z,a1)) = a1 by A29; suppose a1 <> c; then not a1 in {c} by TARSKI:def 1; then A35: a1 in F by A28,XBOOLE_0:def 2; InitSegm(Z,a1) = InitSegm(F,a1) proof thus InitSegm(Z,a1) c= InitSegm(F,a1) proof let x; assume x in InitSegm(Z,a1); then A36: x in LowerCone{a1} /\ Z by ORDERS_1:def 14; then A37: x in LowerCone{a1} by XBOOLE_0:def 3; now assume A38: not x in F; x in Z by A36,XBOOLE_0:def 3; then x in {c} by A38,XBOOLE_0:def 2; then x = c by TARSKI:def 1; then c in {c1 : for c2 st c2 in {a1} holds c1 < c2} by A37,ORDERS_1: def 13; then A39: ex c1 st c1 = c & for c2 st c2 in {a1} holds c1 < c2; a1 in {a1} by TARSKI:def 1; then c < a1 & a1 < c by A5,A35,A39; hence contradiction by ORDERS_1:28; end; then x in LowerCone{a1} /\ F by A37,XBOOLE_0:def 3; hence thesis by ORDERS_1:def 14; end; F c= Z by XBOOLE_1:7; hence thesis by ORDERS_1:65; end; hence f.UpperCone(InitSegm(Z,a1)) = a1 by A35,ORDERS_1:def 16; end; hence f.UpperCone(InitSegm(Z,a1)) = a1; end; then reconsider Z as Chain of f by A26,A27,ORDERS_1:def 16; c in {c} by TARSKI:def 1; then Z in Chains(f) & c in Z by ORDERS_1:def 17,XBOOLE_0:def 2; then c in F by TARSKI:def 4; hence thesis by A5; end; definition let A be non empty set, O be Order of A; cluster RelStr(#A,O#) -> non empty; coherence; end; theorem Th74: (for C ex a st for b st b in C holds a <= b) implies ex a st for b holds not b < a proof assume A1: for C ex a st for b st b in C holds a <= b; set X = the carrier of A; defpred P[Element of A, Element of A] means ex a1,a2 st a1 = $2 & a2 = $1 & a1 <= a2; set R = (the InternalRel of A)~; A2: for a,b being Element of A holds [a,b] in R iff b <= a proof let a,b be Element of A; [a,b] in R iff [b,a] in the InternalRel of A by RELAT_1:def 7; hence [a,b] in R iff b <= a by ORDERS_1:def 9; end; A3: R is reflexive by RELAT_2:27; A4: dom R = dom the InternalRel of A by RELAT_2:29 .= X by PARTFUN1:def 4; A5: R is total by A4,PARTFUN1:def 4; A6: R is antisymmetric by RELAT_2:40; R is transitive by RELAT_2:42; then reconsider R as Order of the carrier of A by A3,A5,A6; set B = RelStr (# the carrier of A, R #); for E being Chain of B ex e being Element of B st for f being Element of B st f in E holds f <= e proof let E be Chain of B; reconsider C = E as Subset of A; the InternalRel of A is_strongly_connected_in C proof let x,y; assume A7: x in C & y in C; then reconsider e = x, f = y as Element of B ; A8: (e <= f or f <= e) & R = the InternalRel of B by A7,ORDERS_1:38; reconsider e1 = e, f1 = f as Element of A ; now per cases by A8,ORDERS_1:def 9; suppose [e,f] in R; then f1 <= e1 by A2; hence thesis by ORDERS_1:def 9; suppose [f,e] in R; then e1 <= f1 by A2; hence thesis by ORDERS_1:def 9; end; hence thesis; end; then reconsider C as Chain of A by ORDERS_1:def 11; consider a such that A9: for b st b in C holds a <= b by A1; reconsider e = a as Element of B; take e; let f be Element of B; reconsider b = f as Element of A; assume f in E; then a <= b by A9; then [f,e] in R & R = the InternalRel of B by A2; hence thesis by ORDERS_1:def 9; end; then consider e being Element of B such that A10: for f being Element of B holds not e < f by Th73; reconsider d = e as Element of A; take d; let b; reconsider f = b as Element of B; assume A11: b < d; then b <= d by ORDERS_1:def 10; then [e,f] in R & R = the InternalRel of B by A2; then e <= f & b <> d by A11,ORDERS_1:def 9; then e < f by ORDERS_1:def 10; hence thesis by A10; end; reserve A,C for Ordinal; theorem Th75: for R,X st R partially_orders X & field R = X & X has_upper_Zorn_property_wrt R ex x st x is_maximal_in R proof let R; let FIELD be set; assume that A1: R is_reflexive_in FIELD & R is_transitive_in FIELD & R is_antisymmetric_in FIELD and A2: field R = FIELD and A3: FIELD has_upper_Zorn_property_wrt R; reconsider XD = FIELD as non empty set by A3,Th51; consider D such that A4: D = XD; not D in {{}} & D in bool D by TARSKI:def 1,ZFMISC_1:def 1; then reconsider M = (bool D) \ {{}} as non empty set by XBOOLE_0:def 4; consider f being Choice_Function of M; {} in {{}} by TARSKI:def 1; then A5: not {} in M by XBOOLE_0:def 4; defpred P[set,set] means $1 <> {} & $2 = f.$1 or $1 = {} & $2 = D; A6: for x st x in bool D ex y st P[x,y] proof let x such that x in bool D; x = {} implies thesis; hence thesis; end; A7: for x,y,z st x in bool D & P[x,y] & P[x,z] holds y = z; consider g such that A8: dom g = bool D & for x st x in bool D holds P[x,g.x] from FuncEx(A7,A6); A9: g.{} = D proof {} c= D by XBOOLE_1:2; hence thesis by A8; end; A10: X in bool D & X <> {} implies g.X in X proof assume A11: X in bool D & X <> {}; then P[X,g.X] & P[X,f.X] & not X in {{}} by A8,TARSKI:def 1; then f.X = g.X & X in M by A11,XBOOLE_0:def 4; hence g.X in X by A5,ORDERS_1:def 1; end; A12: for L holds { d : for x st x in rng L holds x <> d & [x,d] in R } c= D proof let L,x; assume x in { d : for x st x in rng L holds x <> d & [x,d] in R }; then ex d1 st x = d1 & for x st x in rng L holds x <> d1 & [x,d1] in R; hence thesis; end; A13: now let A,B,L1,L2; assume that A14: dom L1 = A & for C,L st C in A & L = L1|C holds L1.C = g.{ d2 : for x st x in rng L holds x <> d2 & [x,d2] in R } and A15: dom L2 = B & for C,L st C in B & L = L2|C holds L2.C = g.{ d2 : for x st x in rng L holds x <> d2 & [x,d2] in R }; let C such that A16: C c= A & C c= B; defpred P[Ordinal] means $1 c= C implies L1|$1 = L2|$1; A17: for A1 st for A2 st A2 in A1 holds P[A2] holds P[A1] proof let A1 such that A18: for A2 st A2 in A1 holds A2 c= C implies L1|A2 = L2|A2 and A19: A1 c= C; A1 c= A & A1 c= B by A16,A19,XBOOLE_1:1; then A20: dom(L1|A1) = A1 & dom(L2|A1) = A1 by A14,A15,RELAT_1:91; now let x; assume A21: x in A1; then reconsider A3 = x as Ordinal by ORDINAL1:23; A3 in C by A19,A21; then A3 in A & A3 in B & A3 c= A1 & A3 c= C by A16,A21,ORDINAL1:def 2; then L1|A3 = L2|A3 & L1.x = L1|A1.x & L2.x = L2|A1.x & L1.A3 = g.{ d1 : for x st x in rng(L1|A3) holds x <> d1 & [x,d1] in R } & L2.A3 = g.{ d2 : for x st x in rng(L2|A3) holds x <> d2 & [x,d2] in R } by A14,A15,A18,A21,FUNCT_1:72; hence L1|A1.x = L2|A1.x; end; hence L1|A1 = L2|A1 by A20,FUNCT_1:9; end; for A1 holds P[A1] from Transfinite_Ind(A17); hence L1|C = L2|C; end; defpred ON[Ordinal,set] means ex L st $2 = g.{ d : for x st x in rng L holds x <> d & [x,d] in R } & dom L = $1 & for A,L1 st A in $1 & L1 = L|A holds L.A = g.{ d1 : for x st x in rng L1 holds x <> d1 & [x,d1] in R }; A22: ON[A,x] & ON[A,y] implies x = y proof given L1 such that A23: x = g.{ d1 : for x st x in rng L1 holds x <> d1 & [x,d1] in R } and A24: dom L1 = A & for C,L st C in A & L = L1|C holds L1.C = g.{ d2 : for x st x in rng L holds x <> d2 & [x,d2] in R }; given L2 such that A25: y = g.{ d1 : for x st x in rng L2 holds x <> d1 & [x,d1] in R } and A26: dom L2 = A & for C,L st C in A & L = L2|C holds L2.C = g.{ d2 : for x st x in rng L holds x <> d2 & [x,d2] in R }; defpred P[Ordinal] means $1 c= A implies L1|$1 = L2|$1; A27: for A1 st for A2 st A2 in A1 holds P[A2] holds P[A1] proof let A1 such that A28: for A2 st A2 in A1 holds A2 c= A implies L1|A2 = L2|A2 and A29: A1 c= A; A30: dom(L1|A1) = A1 & dom(L2|A1) = A1 by A24,A26,A29,RELAT_1:91; now let x; assume A31: x in A1; then reconsider A3 = x as Ordinal by ORDINAL1:23; A3 in A & A3 c= A1 & A3 c= A by A29,A31,ORDINAL1:def 2; then L1|A3 = L2|A3 & L1.x = L1|A1.x & L2.x = L2|A1.x & L1.A3 = g.{ d1 : for x st x in rng(L1|A3) holds x <> d1 & [x,d1] in R } & L2.A3 = g.{ d2 : for x st x in rng(L2|A3) holds x <> d2 & [x,d2] in R } by A24,A26,A28,A31,FUNCT_1:72; hence L1|A1.x = L2|A1.x; end; hence L1|A1 = L2|A1 by A30,FUNCT_1:9; end; for A1 holds P[A1] from Transfinite_Ind(A27); then L1|A = L2|A & L1 = L1|A & L2 = L2|A by A24,A26,RELAT_1:97; hence x = y by A23,A25; end; A32: for d,A,B st ON[A,d] & ON[B,d] holds A = B proof let d,A,B; given L1 such that A33: d = g.{ d1 : for x st x in rng L1 holds x <> d1 & [x,d1] in R } and A34: dom L1 = A & for C,L st C in A & L = L1|C holds L1.C = g.{ d2 : for x st x in rng L holds x <> d2 & [x,d2] in R }; given L2 such that A35: d = g.{ d1 : for x st x in rng L2 holds x <> d1 & [x,d1] in R } and A36: dom L2 = B & for C,L st C in B & L = L2|C holds L2.C = g.{ d2 : for x st x in rng L holds x <> d2 & [x,d2] in R }; A37: now set X = { d1 : for x st x in rng(L2|A) holds x <> d1 & [x,d1] in R }; set Y = { d1 : for x st x in rng L2 holds x <> d1 & [x,d1] in R }; assume A38: A in B; then A39: L2.A = g.X & A c= B & A c= A by A36,ORDINAL1:def 2; then L2|A = L1|A by A13,A34,A36 .= L1 by A34,RELAT_1:97; then A40: d in rng L2 by A33,A36,A38,A39,FUNCT_1:def 5; A41: Y c= D proof let x; assume x in Y; then ex d1 st x = d1 & for x st x in rng L2 holds x <> d1 & [x,d1] in R; hence thesis; end; then A42: P[Y,g.Y] by A8; A43: not d in d; now assume Y <> {}; then not Y in {{}} & P[Y,f.Y] by TARSKI:def 1; then Y in M & g.Y = f.Y by A8,A41,XBOOLE_0:def 4; then d in Y by A5,A35,ORDERS_1:def 1; then ex d1 st d = d1 & for x st x in rng L2 holds x <> d1 & [x,d1] in R; hence contradiction by A40; end; hence contradiction by A35,A42,A43; end; now set X = { d1 : for x st x in rng(L1|B) holds x <> d1 & [x,d1] in R }; set Y = { d1 : for x st x in rng L1 holds x <> d1 & [x,d1] in R }; assume A44: B in A; then A45: L1.B = g.X & B c= A & B c= B by A34,ORDINAL1:def 2; then L1|B = L2|B by A13,A34,A36 .= L2 by A36,RELAT_1:97; then A46: d in rng L1 by A34,A35,A44,A45,FUNCT_1:def 5; A47: Y c= D proof let x; assume x in Y; then ex d1 st x = d1 & for x st x in rng L1 holds x <> d1 & [x,d1] in R; hence thesis; end; then A48: P[Y,g.Y] by A8; A49: not d in d; now assume Y <> {}; then not Y in {{}} & P[Y,f.Y] by TARSKI:def 1; then Y in M & g.Y = f.Y by A8,A47,XBOOLE_0:def 4; then d in Y by A5,A33,ORDERS_1:def 1; then ex d1 st d = d1 & for x st x in rng L1 holds x <> d1 & [x,d1] in R; hence contradiction by A46; end; hence contradiction by A33,A48,A49; end; hence thesis by A37,ORDINAL1:24; end; defpred OM[Ordinal] means ON[$1,D]; A50: ex A st OM[A] proof assume A51: for A holds not ON[A,D]; defpred OO[Ordinal] means ex d st ON[$1,d]; A52: for A st for B st B in A holds OO[B] holds OO[A] proof let A such that A53: for B st B in A ex d st ON[B,d]; defpred P[set,set] means ex C st $1 = C & ON[C,$2]; A54: for x,y,z st x in A & P[x,y] & P[x,z] holds y = z by A22; A55: for x st x in A ex y st P[x,y] proof let x; assume A56: x in A; then reconsider C = x as Ordinal by ORDINAL1:23; consider d such that A57: ON[C,d] by A53,A56; reconsider y = d as set; take y,C; thus thesis by A57; end; consider h such that A58: dom h = A & for x st x in A holds P[x,h.x] from FuncEx(A54,A55); reconsider h as T-Sequence by A58,ORDINAL1:def 7; set X = { d1 : for x st x in rng h holds x <> d1 & [x,d1] in R }; A59: ON[A,g.X] proof take h; thus g.X = g.{ d1 : for x st x in rng h holds x <> d1 & [x,d1] in R }& dom h = A by A58; let B,L; assume A60: B in A & L = h|B; then ex C st B = C & ON[C,h.B] by A58; then consider L1 such that A61: h.B = g.{d1: for x st x in rng L1 holds x <> d1 & [x,d1] in R } and A62: dom L1 = B & for C,L st C in B & L = L1|C holds L1.C = g.{ d1 : for x st x in rng L holds x <> d1 & [x,d1] in R }; A63: for x st x in B holds L1.x = h|B.x proof let x; assume A64: x in B; then reconsider A1 = x as Ordinal by ORDINAL1:23; A65: ON[A1,L1.x] proof reconsider K = L1|A1 as T-Sequence; take K; thus L1.x = g.{d1: for x st x in rng K holds x <> d1 & [x,d1] in R } by A62,A64; A1 c= B by A64,ORDINAL1:def 2; hence dom K = A1 by A62,RELAT_1:91; let A2,L2; assume A66: A2 in A1 & L2 = K|A2; then A67: A2 c= A1 & A2 in B by A64,ORDINAL1:19,def 2; then L2 = L1|A2 & K.A2 = L1.A2 by A66,FUNCT_1:72,82; hence thesis by A62,A67; end; A1 in A by A60,A64,ORDINAL1:19; then A68: ex A2 st x = A2 & ON[A2,h.x] by A58; h|B.x = h.x by A64,FUNCT_1:72; hence thesis by A22,A65,A68; end; B c= A by A60,ORDINAL1:def 2; then dom(h|B) = B by A58,RELAT_1:91; then h|B = L1 by A62,A63,FUNCT_1:9; hence thesis by A60,A61; end; then A69: X <> {} by A9,A51; A70: X c= D proof let x; assume x in X; then ex d1 st x = d1 & for x st x in rng h holds x <> d1 & [x,d1] in R; hence thesis; end; then g.X in X by A10,A69; then reconsider d = g.X as Element of D by A70; take d; thus thesis by A59; end; A71: for A holds OO[A] from Transfinite_Ind(A52); defpred P[set,set] means ex A st $2 = A & ON[A,$1]; A72: for x,y,z st P[x,y] & P[x,z] holds y = z proof let x,y,z; given A1 such that A73: y = A1 & ON[A1,x]; given A2 such that A74: z = A2 & ON[A2,x]; consider d1 such that A75: ON[A1,d1] by A71; consider d2 such that A76: ON[A2,d2] by A71; d1 = x & d2 = x by A22,A73,A74,A75,A76; hence thesis by A32,A73,A74; end; consider X such that A77: x in X iff ex y st y in D & P[y,x] from Fraenkel(A72); for A holds A in X proof let A; ex d st ON[A,d] by A71; hence thesis by A77; end; hence contradiction by ORDINAL1:38; end; consider A such that A78: OM[A] & for B st OM[B] holds A c= B from Ordinal_Min(A50); defpred P[set] means ex B st B in A & ON[B,$1]; consider X such that A79: x in X iff x in D & P[x] from Separation; D <> {} & D in bool D by ZFMISC_1:def 1; then reconsider d = g.D as Element of D by A10; A80: d in D & ON[{},d] proof thus d in D; deffunc H(Ordinal) = 0; consider L such that A81: dom L = {} & for B,L1 st B in {} & L1 = L|B holds L.B = H(L1) from TS_Exist; take L; A82: { d1 : for x st x in rng L holds x <> d1 & [x,d1] in R } c= D by A12; D c= { d1 : for x st x in rng L holds x <> d1 & [x,d1] in R } proof let x; assume x in D; then reconsider d = x as Element of D; for x st x in rng L holds x <> d & [x,d] in R by A81,RELAT_1:65; hence thesis; end; hence d = g.{ d1 : for x st x in rng L holds x <> d1 & [x,d1] in R } by A82,XBOOLE_0:def 10; thus dom L = {} by A81; thus thesis; end; not Y in Y; then d <> D; then {} c= A & {} <> A by A22,A78,A80,XBOOLE_1:2; then {} c< A by XBOOLE_0:def 8; then {} in A by ORDINAL1:21; then reconsider X as non empty set by A79,A80; A83: X c= D proof let x; thus thesis by A79; end; R|_2 X is_linear-order proof R is reflexive by A1,A2,RELAT_2:def 9; hence R|_2 X is reflexive by WELLORD1:22; R is transitive by A1,A2,RELAT_2:def 16; hence R|_2 X is transitive by WELLORD1:24; R is antisymmetric by A1,A2,RELAT_2:def 12; hence R|_2 X is antisymmetric by WELLORD1:25; thus R|_2 X is connected proof let x,y; assume x in field(R|_2 X) & y in field(R|_2 X); then A84: x in X & y in X by WELLORD1:19; then reconsider x' = x, y' = y as Element of D by A79; A85: [x,y] in [:X,X:] & [y,x] in [:X,X:] by A84,ZFMISC_1:106; consider A1 such that A86: A1 in A & ON[A1,x] by A79,A84; consider A2 such that A87: A2 in A & ON[A2,y] by A79,A84; consider L1 such that A88: x' = g.{ d1 : for x st x in rng L1 holds x <> d1 & [x,d1] in R } and A89: dom L1 = A1 & for C,L st C in A1 & L = L1|C holds L1.C = g.{ d2 : for x st x in rng L holds x <> d2 & [x,d2] in R } by A86; consider L2 such that A90: y' = g.{ d1 : for x st x in rng L2 holds x <> d1 & [x,d1] in R } and A91: dom L2 = A2 & for C,L st C in A2 & L = L2|C holds L2.C = g.{ d2 : for x st x in rng L holds x <> d2 & [x,d2] in R } by A87; A92: now set Z = { d1 : for x st x in rng(L2|A1) holds x <> d1 & [x,d1] in R }; set Y = { d1 : for x st x in rng L2 holds x <> d1 & [x,d1] in R }; assume A93: A1 in A2; then A94: L2.A1 = g.Z & A1 c= A2 & A1 c= A1 by A91,ORDINAL1:def 2; then L2|A1 = L1|A1 by A13,A89,A91 .= L1 by A89,RELAT_1:97; then A95: x' in rng L2 by A88,A91,A93,A94,FUNCT_1:def 5; A96: Y c= D by A12; not y' in y'; then Y <> {} by A9,A90; then y' in Y by A10,A90,A96; then ex d1 st y' = d1 & for x st x in rng L2 holds x <> d1 & [x,d1] in R; then [x,y] in R by A95; hence thesis by A85,WELLORD1:16; end; A97: now set Z = { d1 : for x st x in rng(L1|A2) holds x <> d1 & [x,d1] in R }; set Y = { d1 : for x st x in rng L1 holds x <> d1 & [x,d1] in R }; assume A98: A2 in A1; then A99: L1.A2 = g.Z & A2 c= A1 & A2 c= A2 by A89,ORDINAL1:def 2; then L1|A2 = L2|A2 by A13,A89,A91 .= L2 by A91,RELAT_1:97; then A100: y' in rng L1 by A89,A90,A98,A99,FUNCT_1:def 5; A101: Y c= D by A12; not d1 in d1; then Y <> {} by A9,A88; then x' in Y by A10,A88,A101; then ex d1 st x' = d1 & for x st x in rng L1 holds x <> d1 & [x,d1] in R; then [y,x] in R by A100; hence thesis by A85,WELLORD1:16; end; A1 = A2 implies thesis by A22,A86,A87; hence thesis by A92,A97,ORDINAL1:24; end; end; then consider x such that A102: x in D & for y st y in X holds [y,x] in R by A3,A4,A83,Def7; take x; thus x in field R by A2,A4,A102; let y such that A103: y in field R & y <> x & [x,y] in R; consider L such that A104: D = g.{ d1 : for x st x in rng L holds x <> d1 & [x,d1] in R } and A105: dom L = A & for B,L1 st B in A & L1 = L|B holds L.B = g.{ d2 : for x st x in rng L1 holds x <> d2 & [x,d2] in R } by A78; A106: rng L c= X proof let z; assume z in rng L; then consider y such that A107: y in dom L & z = L.y by FUNCT_1:def 5; reconsider y as Ordinal by A107,ORDINAL1:23; A108: ON[y,z] proof reconsider K = L|y as T-Sequence; take K; thus z = g.{ d2 : for x st x in rng K holds x <> d2 & [x,d2] in R } by A105,A107; y c= dom L by A107,ORDINAL1:def 2; hence A109: dom K = y by RELAT_1:91; let B,L1; assume A110: B in y & L1 = K|B; then A111: K.B = L.B by A109,FUNCT_1:70; B c= y by A110,ORDINAL1:def 2; then L1 = L|B & B in A by A105,A107,A110,FUNCT_1:82,ORDINAL1:19; hence K.B = g.{ d1 : for x st x in rng L1 holds x <> d1 & [x,d1] in R } by A105,A111; end; set Z = { d2 : for x st x in rng(L|y) holds x <> d2 & [x,d2] in R }; A112: now assume Z = {}; then z = D by A9,A105,A107; then dom L c= y by A78,A105,A108; hence contradiction by A107,ORDINAL1:7; end; A113: Z c= D by A12; then g.Z in Z by A10,A112; then z in Z by A105,A107; hence z in X by A79,A105,A107,A108,A113; end; set Z = { d1 : for x st x in rng L holds x <> d1 & [x,d1] in R }; A114: now assume y in X; then [y,x] in R & R is_antisymmetric_in field R by A1,A2,A102; hence contradiction by A2,A4,A102,A103,RELAT_2:def 4; end; reconsider y' = y as Element of D by A2,A4,A103; now let z; assume A115: z in rng L; hence z <> y' by A106,A114; reconsider z' = z as Element of X by A106,A115; [z',x] in R & R is_transitive_in field R & z in D by A1,A2,A79,A102,A106,A115; hence [z,y] in R by A2,A4,A102,A103,RELAT_2:def 8; end; then A116: y in Z & Z c= D by A12; then D in Z by A10,A104; hence contradiction by A116,ORDINAL1:7; end; theorem for R,X st R partially_orders X & field R = X & X has_lower_Zorn_property_wrt R ex x st x is_minimal_in R proof let R,X such that A1: R partially_orders X & field R = X & X has_lower_Zorn_property_wrt R; R = R~~; then X has_upper_Zorn_property_wrt R~ & R~ partially_orders X & field(R~) = X by A1,Th39,Th53,RELAT_1:38; then consider x such that A2: x is_maximal_in R~ by Th75; take x; thus thesis by A2,Th65; end; theorem Th77: for X st X <> {} & for Z st Z c= X & Z is c=-linear ex Y st Y in X & for X1 st X1 in Z holds X1 c= Y ex Y st Y in X & for Z st Z in X & Z <> Y holds not Y c= Z proof let X; assume that A1: X <> {} and A2: for Z st Z c= X & Z is c=-linear ex Y st Y in X & for X1 st X1 in Z holds X1 c= Y; reconsider D = X as non empty set by A1; set R = RelIncl D; A3: field R = D by WELLORD2:def 1; A4: R partially_orders D proof R is reflexive & R is transitive & R is antisymmetric by WELLORD2:2,3,5; hence R is_reflexive_in D & R is_transitive_in D & R is_antisymmetric_in D by A3,RELAT_2:def 9,def 12,def 16; end; D has_upper_Zorn_property_wrt R proof let Z; assume A5: Z c= D & R|_2 Z is_linear-order; then R|_2 Z is connected by Def3; then A6: R|_2 Z is_connected_in field(R|_2 Z) by RELAT_2:def 14; A7: Z c= field(R|_2 Z) proof let x; assume x in Z; then [x,x] in [:Z,Z:] & [x,x] in R by A5,WELLORD2:def 1,ZFMISC_1:106 ; then [x,x] in R|_2 Z by WELLORD1:16; hence thesis by RELAT_1:30; end; set Q = R|_2 Z; Z is c=-linear proof let X1,X2; assume A8: X1 in Z & X2 in Z; then X1 <> X2 implies [X1,X2] in Q or [X2,X1] in Q by A6,A7,RELAT_2: def 6 ; then (X1 <> X2 implies [X1,X2] in R or [X2,X1] in R) & X1 in D & X2 in D & X1 c= X1 by A5,A8,WELLORD1:16; hence X1 c= X2 or X2 c= X1 by WELLORD2:def 1; end; then consider Y such that A9: Y in X & for X1 st X1 in Z holds X1 c= Y by A2,A5; take x = Y; thus x in D by A9; let y; assume y in Z; then y c= Y & y in D by A5,A9; hence thesis by A9,WELLORD2:def 1; end; then consider x such that A10: x is_maximal_in R by A3,A4,Th75; take Y = x; A11: Y in field R & not ex y st y in field R & y <> Y & [Y,y] in R by A10,Def9; thus Y in X by A3,A10,Def9; let Z; assume A12: Z in X & Z <> Y; then not [Y,Z] in R by A3,A10,Def9; hence not Y c= Z by A3,A11,A12,WELLORD2:def 1; end; theorem Th78: for X st X <> {} & for Z st Z c= X & Z is c=-linear ex Y st Y in X & for X1 st X1 in Z holds Y c= X1 ex Y st Y in X & for Z st Z in X & Z <> Y holds not Z c= Y proof let X; assume that A1: X <> {} and A2: for Z st Z c= X & Z is c=-linear ex Y st Y in X & for X1 st X1 in Z holds Y c= X1; reconsider D = X as non empty set by A1; set R = (RelIncl D)~; A3: field R = field RelIncl D by RELAT_1:38 .= D by WELLORD2:def 1; A4: x in D implies [x,x] in R proof x in D implies [x,x] in RelIncl D by WELLORD2:def 1; hence thesis by RELAT_1:def 7; end; A5: R partially_orders D proof RelIncl D is reflexive & RelIncl D is transitive & RelIncl D is antisymmetric & field RelIncl D = D by WELLORD2:2,3,5,def 1; then RelIncl D is_reflexive_in D & RelIncl D is_transitive_in D & RelIncl D is_antisymmetric_in D by RELAT_2:def 9,def 12,def 16; hence R is_reflexive_in D & R is_transitive_in D & R is_antisymmetric_in D by Lm9,Lm10,Lm11; end; D has_upper_Zorn_property_wrt R proof let Z; assume A6: Z c= D & R|_2 Z is_linear-order; then R|_2 Z is connected by Def3; then A7: R|_2 Z is_connected_in field(R|_2 Z) by RELAT_2:def 14; A8: Z c= field(R|_2 Z) proof let x; assume x in Z; then [x,x] in [:Z,Z:] & [x,x] in R by A4,A6,ZFMISC_1:106; then [x,x] in R|_2 Z by WELLORD1:16; hence thesis by RELAT_1:30; end; set Q = R|_2 Z; Z is c=-linear proof let X1,X2; assume A9: X1 in Z & X2 in Z; then X1 <> X2 implies [X1,X2] in Q or [X2,X1] in Q by A7,A8,RELAT_2: def 6 ; then (X1 <> X2 implies [X1,X2] in R or [X2,X1] in R) & X1 in D & X2 in D by A6,A9,WELLORD1:16; then (X1 <> X2 implies [X1,X2] in RelIncl D or [X2,X1] in RelIncl D) & X1 in D & X2 in D & X1 c= X1 by RELAT_1:def 7; hence X1 c= X2 or X2 c= X1 by WELLORD2:def 1; end; then consider Y such that A10: Y in X & for X1 st X1 in Z holds Y c= X1 by A2,A6; take x = Y; thus x in D by A10; let y; assume y in Z; then Y c= y & y in D by A6,A10; then [Y,y] in RelIncl D by A10,WELLORD2:def 1; hence thesis by RELAT_1:def 7; end; then consider x such that A11: x is_maximal_in R by A3,A5,Th75; take Y = x; A12: Y in field R & not ex y st y in field R & y <> Y & [Y,y] in R by A11,Def9; thus Y in X by A3,A11,Def9; let Z; assume A13: Z in X & Z <> Y; then not [Y,Z] in R by A3,A11,Def9; then not [Z,Y] in RelIncl D by RELAT_1:def 7; hence not Z c= Y by A3,A12,A13,WELLORD2:def 1; end; theorem Th79: for X st X <> {} & for Z st Z <> {} & Z c= X & Z is c=-linear holds union Z in X ex Y st Y in X & for Z st Z in X & Z <> Y holds not Y c= Z proof let X such that A1: X <> {} and A2: for Z st Z <> {} & Z c= X & Z is c=-linear holds union Z in X; for Z st Z c= X & Z is c=-linear ex Y st Y in X & for X1 st X1 in Z holds X1 c= Y proof let Z such that A3: Z c= X & Z is c=-linear; consider x being Element of X; Z <> {} or Z = {}; then consider Y such that A4: Y = union Z & Z <> {} or Y = x & Z = {}; take Y; thus thesis by A1,A2,A3,A4,ZFMISC_1:92; end; hence thesis by A1,Th77; end; theorem for X st X <> {} & for Z st Z <> {} & Z c= X & Z is c=-linear holds meet Z in X ex Y st Y in X & for Z st Z in X & Z <> Y holds not Z c= Y proof let X such that A1: X <> {} and A2: for Z st Z <> {} & Z c= X & Z is c=-linear holds meet Z in X; for Z st Z c= X & Z is c=-linear ex Y st Y in X & for X1 st X1 in Z holds Y c= X1 proof let Z such that A3: Z c= X & Z is c=-linear; consider x being Element of X; Z <> {} or Z = {}; then consider Y such that A4: Y = meet Z & Z <> {} or Y = x & Z = {}; take Y; thus thesis by A1,A2,A3,A4,SETFAM_1:4; end; hence thesis by A1,Th78; end; scheme Zorn_Max{A() -> non empty set, P[set,set]}: ex x being Element of A() st for y being Element of A() st x <> y holds not P[x,y] provided A1: for x being Element of A() holds P[x,x] and A2: for x,y being Element of A() st P[x,y] & P[y,x] holds x = y and A3: for x,y,z being Element of A() st P[x,y] & P[y,z] holds P[x,z] and A4: for X st X c= A() & (for x,y being Element of A() st x in X & y in X holds P[x,y] or P[y,x]) holds ex y being Element of A() st for x being Element of A() st x in X holds P[x,y] proof defpred PP[set,set] means P[$1,$2]; consider R being Relation of A() such that A5: for x,y being Element of A() holds [x,y] in R iff PP[x,y] from Rel_On_Dom_Ex; A6: A() c= dom R proof let x; assume x in A(); then reconsider x' = x as Element of A(); P[x',x'] by A1; then [x,x] in R by A5; hence thesis by RELAT_1:def 4; end; A7: dom R = A() by A6,XBOOLE_0:def 10; then A8: R is total by PARTFUN1:def 4; A9: field R = dom R \/ rng R by RELAT_1:def 6; then A10: field R c= A(); A() c= field R by A7,A9,XBOOLE_1:7; then A11: field R = A() by A10,XBOOLE_0:def 10; R is_reflexive_in A() proof let x; assume x in A(); then reconsider x' = x as Element of A(); P[x',x'] by A1; hence thesis by A5; end; then A12: R is reflexive by A11,RELAT_2:def 9; R is_antisymmetric_in A() proof let x,y; assume x in A() & y in A(); then reconsider x' = x, y' = y as Element of A(); assume [x,y] in R & [y,x] in R; then P[x',y'] & P[y',x'] by A5; hence thesis by A2; end; then A13: R is antisymmetric by A11,RELAT_2:def 12; R is_transitive_in A() proof let x,y,z; assume x in A() & y in A() & z in A(); then reconsider x' = x, y' = y, z' = z as Element of A(); assume [x,y] in R & [y,z] in R; then P[x',y'] & P[y',z'] by A5; then P[x',z'] by A3; hence thesis by A5; end; then R is transitive by A11,RELAT_2:def 16; then reconsider R as Order of A() by A8,A12,A13; set B = RelStr (# A(),R #); now let C be Chain of B; now let x,y be Element of A(); reconsider a = x, b = y as Element of B; assume x in C & y in C; then a <= b or b <= a by ORDERS_1:38; then [a,b] in R or [b,a] in R by ORDERS_1:def 9; hence P[x,y] or P[y,x] by A5; end; then consider y being Element of A() such that A14: for x being Element of A() st x in C holds P[x,y] by A4; reconsider a' = y as Element of B; take a = a'; let b be Element of B; reconsider x = b as Element of A(); assume b in C; then P[x,y] by A14; then [x,y] in R by A5; hence b <= a by ORDERS_1:def 9; end; then consider a being Element of B such that A15: for b being Element of B holds not a < b by Th73; reconsider x = a as Element of A(); take x; let y be Element of A(); reconsider b = y as Element of B; assume A16: x <> y; not a < b by A15; then not a <= b by A16,ORDERS_1:def 10; then not [x,y] in R by ORDERS_1:def 9; hence thesis by A5; end; scheme Zorn_Min{A() -> non empty set, P[set,set]}: ex x being Element of A() st for y being Element of A() st x <> y holds not P[y,x] provided A1: for x being Element of A() holds P[x,x] and A2: for x,y being Element of A() st P[x,y] & P[y,x] holds x = y and A3: for x,y,z being Element of A() st P[x,y] & P[y,z] holds P[x,z] and A4: for X st X c= A() & (for x,y being Element of A() st x in X & y in X holds P[x,y] or P[y,x]) holds ex y being Element of A() st for x being Element of A() st x in X holds P[y,x] proof defpred PP[set,set] means P[$1,$2]; consider R being Relation of A() such that A5: for x,y being Element of A() holds [x,y] in R iff PP[x,y] from Rel_On_Dom_Ex; A6: A() c= dom R proof let x; assume x in A(); then reconsider x' = x as Element of A(); P[x',x'] by A1; then [x,x] in R by A5; hence thesis by RELAT_1:def 4; end; A7: dom R = A() by A6,XBOOLE_0:def 10; then A8: R is total by PARTFUN1:def 4; A9: field R = dom R \/ rng R by RELAT_1:def 6; then A10: field R c= A(); A() c= field R by A7,A9,XBOOLE_1:7; then A11: field R = A() by A10,XBOOLE_0:def 10; R is_reflexive_in A() proof let x; assume x in A(); then reconsider x' = x as Element of A(); P[x',x'] by A1; hence thesis by A5; end; then A12: R is reflexive by A11,RELAT_2:def 9; R is_antisymmetric_in A() proof let x,y; assume x in A() & y in A(); then reconsider x' = x, y' = y as Element of A(); assume [x,y] in R & [y,x] in R; then P[x',y'] & P[y',x'] by A5; hence thesis by A2; end; then A13: R is antisymmetric by A11,RELAT_2:def 12; R is_transitive_in A() proof let x,y,z; assume x in A() & y in A() & z in A(); then reconsider x' = x, y' = y, z' = z as Element of A(); assume [x,y] in R & [y,z] in R; then P[x',y'] & P[y',z'] by A5; then P[x',z'] by A3; hence thesis by A5; end; then R is transitive by A11,RELAT_2:def 16; then reconsider R as Order of A() by A8,A12,A13; reconsider R as Order of A(); set B = RelStr (# A(),R #); now let C be Chain of B; now let x,y be Element of A(); reconsider a = x, b = y as Element of B; assume x in C & y in C; then a <= b or b <= a by ORDERS_1:38; then [a,b] in R or [b,a] in R by ORDERS_1:def 9; hence P[x,y] or P[y,x] by A5; end; then consider y being Element of A() such that A14: for x being Element of A() st x in C holds P[y,x] by A4; reconsider a' = y as Element of B; take a = a'; let b be Element of B; reconsider x = b as Element of A(); assume b in C; then P[y,x] by A14; then [y,x] in R by A5; hence a <= b by ORDERS_1:def 9; end; then consider a being Element of B such that A15: for b being Element of B holds not b < a by Th74; reconsider x = a as Element of A(); take x; let y be Element of A(); reconsider b = y as Element of B; assume A16: x <> y; not b < a by A15; then not b <= a by A16,ORDERS_1:def 10; then not [y,x] in R by ORDERS_1:def 9; hence thesis by A5; end; :: :: Orders - continuation. :: theorem R partially_orders X & field R = X implies ex P st R c= P & P linearly_orders X & field P = X proof assume A1: R partially_orders X & field R = X; defpred P[set] means ex P st $1 = P & R c= P & P partially_orders X & field P = X; consider Rel being set such that A2: x in Rel iff x in bool [:X,X:] & P[x] from Separation; R c= [:X,X:] by A1,Lm2; then A3: Rel <> {} by A1,A2; for Z st Z <> {} & Z c= Rel & Z is c=-linear holds union Z in Rel proof let Z; assume that A4: Z <> {} & Z c= Rel and A5: Z is c=-linear; reconsider T = [:X,X:] as Relation by RELAT_1:6; Z c= bool [:X,X:] proof let x; assume x in Z; hence thesis by A2,A4; end; then A6: union Z c= union bool [:X,X:] & union bool [:X,X:] = T by ZFMISC_1:95,99; then reconsider S = union Z as Relation by RELAT_1:3; consider y being Element of Z; y in Rel by A4,TARSKI:def 3; then consider P such that A7: y = P & R c= P & P partially_orders X & field P = X by A2; A8: P is_reflexive_in X by A7,Def5; A9: R c= S proof let x,y; assume [x,y] in R; hence thesis by A4,A7,TARSKI:def 4; end; A10: field S = X proof thus field S c= X proof let x; assume x in field S; then A11: x in dom S \/ rng S by RELAT_1:def 6; A12: now assume x in dom S; then consider y such that A13: [x,y] in S by RELAT_1:def 4; consider Y such that A14: [x,y] in Y & Y in Z by A13,TARSKI:def 4; ex P st Y = P & R c= P & P partially_orders X & field P = X by A2,A4,A14; hence thesis by A14,RELAT_1:30; end; now assume x in rng S; then consider y such that A15: [y,x] in S by RELAT_1:def 5; consider Y such that A16: [y,x] in Y & Y in Z by A15,TARSKI:def 4; ex P st Y = P & R c= P & P partially_orders X & field P = X by A2,A4,A16; hence thesis by A16,RELAT_1:30; end; hence thesis by A11,A12,XBOOLE_0:def 2; end; thus X c= field S by A1,A9,RELAT_1:31; end; S partially_orders X proof thus S is_reflexive_in X proof let x; assume x in X; then [x,x] in P by A8,RELAT_2:def 1; hence thesis by A4,A7,TARSKI:def 4; end; thus S is_transitive_in X proof let x,y,z; assume A17: x in X & y in X & z in X & [x,y] in S & [y,z] in S; then consider X1 such that A18: [x,y] in X1 & X1 in Z by TARSKI:def 4; consider X2 such that A19: [y,z] in X2 & X2 in Z by A17,TARSKI:def 4; X1,X2 are_c=-comparable by A5,A18,A19,ORDINAL1:def 9; then A20: X1 c= X2 or X2 c= X1 by XBOOLE_0:def 9; consider P1 being Relation such that A21: X1 = P1 & R c= P1 & P1 partially_orders X & field P1 = X by A2,A4 ,A18; consider P2 being Relation such that A22: X2 = P2 & R c= P2 & P2 partially_orders X & field P2 = X by A2,A4 ,A19; P1 is_transitive_in X & P2 is_transitive_in X by A21,A22,Def5; then [x,z] in P1 or [x,z] in P2 by A17,A18,A19,A20,A21,A22,RELAT_2:def 8; hence thesis by A18,A19,A21,A22,TARSKI:def 4; end; thus S is_antisymmetric_in X proof let x,y; assume A23: x in X & y in X & [x,y] in S & [y,x] in S; then consider X1 such that A24: [x,y] in X1 & X1 in Z by TARSKI:def 4; consider X2 such that A25: [y,x] in X2 & X2 in Z by A23,TARSKI:def 4; X1,X2 are_c=-comparable by A5,A24,A25,ORDINAL1:def 9; then A26: X1 c= X2 or X2 c= X1 by XBOOLE_0:def 9; consider P1 being Relation such that A27: X1 = P1 & R c= P1 & P1 partially_orders X & field P1 = X by A2,A4 ,A24; consider P2 being Relation such that A28: X2 = P2 & R c= P2 & P2 partially_orders X & field P2 = X by A2,A4 ,A25; P1 is_antisymmetric_in X & P2 is_antisymmetric_in X by A27,A28,Def5 ; hence thesis by A23,A24,A25,A26,A27,A28,RELAT_2:def 4; end; end; hence thesis by A2,A6,A9,A10; end; then consider Y such that A29: Y in Rel & for Z st Z in Rel & Z <> Y holds not Y c= Z by A3,Th79; consider P such that A30: Y = P & R c= P & P partially_orders X & field P = X by A2,A29; take P; thus R c= P by A30; thus A31: P is_reflexive_in X & P is_transitive_in X & P is_antisymmetric_in X by A30,Def5; thus P is_connected_in X proof let x,y such that A32: x in X & y in X & x <> y & not [x,y] in P & not [y,x] in P; defpred Q[set,set] means [$1,$2] in P or [$1,x] in P & [y,$2] in P; consider Q being Relation such that A33: [z,u] in Q iff z in X & u in X & Q[z,u] from Rel_Existence; A34: P c= Q proof let z,u; assume A35: [z,u] in P; then z in field P & u in field P by RELAT_1:30; hence thesis by A30,A33,A35; end; then A36: R c= Q by A30,XBOOLE_1:1; A37: Q c= [:X,X:] proof let x; assume A38: x in Q; then consider y,z such that A39: x = [y,z] by RELAT_1:def 1; y in X & z in X by A33,A38,A39; hence thesis by A39,ZFMISC_1:106; end; A40: field Q = X proof thus field Q c= X proof let z; assume z in field Q; then A41: z in dom Q \/ rng Q by RELAT_1:def 6; A42: now assume z in dom Q; then ex u st [z,u] in Q by RELAT_1:def 4; hence z in X by A33; end; now assume z in rng Q; then ex u st [u,z] in Q by RELAT_1:def 5; hence z in X by A33; end; hence thesis by A41,A42,XBOOLE_0:def 2; end; thus X c= field Q by A30,A34,RELAT_1:31; end; Q partially_orders X proof thus Q is_reflexive_in X proof let z; assume A43: z in X; then [z,z] in P by A31,RELAT_2:def 1; hence [z,z] in Q by A33,A43; end; thus Q is_transitive_in X proof let a,b,c be set; assume A44: a in X & b in X & c in X & [a,b] in Q & [b,c] in Q; then ([a,b] in P or [a,x] in P & [y,b] in P) & ([b,c] in P or [b,x] in P & [y,c] in P) by A33; then [a,c] in P or [a,x] in P & [y,c] in P or [a,x] in P & [y,c] in P or [y,x] in P by A31,A32,A44,RELAT_2:def 8; hence [a,c] in Q by A32,A33,A44; end; thus Q is_antisymmetric_in X proof let a,b be set; assume A45: a in X & b in X & [a,b] in Q & [b,a] in Q; then ([a,b] in P or [a,x] in P & [y,b] in P) & ([b,a] in P or [b,x] in P & [y,a] in P) by A33; then a = b or [a,x] in P & [y,a] in P or [y,x] in P by A31,A32,A45,RELAT_2:def 4,def 8; hence a = b by A31,A32,A45,RELAT_2:def 8; end; end; then Q in Rel by A2,A36,A37,A40; then A46: Q = P by A29,A30,A34; [x,x] in P & [y,y] in P & x in X & y in X by A31,A32,RELAT_2:def 1; hence contradiction by A32,A33,A46; end; thus field P = X by A30; end; :: :: Auxiliary theorems. :: theorem R c= [:field R,field R:] by Lm2; theorem R is reflexive & X c= field R implies field(R|_2 X) = X by Lm3; theorem R is_reflexive_in X implies R|_2 X is reflexive by Lm4; theorem R is_transitive_in X implies R|_2 X is transitive by Lm5; theorem R is_antisymmetric_in X implies R|_2 X is antisymmetric by Lm6; theorem R is_connected_in X implies R|_2 X is connected by Lm7; theorem R is_connected_in X & Y c= X implies R is_connected_in Y by Lm8; theorem R well_orders X & Y c= X implies R well_orders Y by Lm15; theorem R is connected implies R~ is connected by Lm1; theorem R is_reflexive_in X implies R~ is_reflexive_in X by Lm9; theorem R is_transitive_in X implies R~ is_transitive_in X by Lm10; theorem R is_antisymmetric_in X implies R~ is_antisymmetric_in X by Lm11; theorem R is_connected_in X implies R~ is_connected_in X by Lm12; theorem (R|_2 X)~ = R~|_2 X by Lm13; theorem R|_2 {} = {} by Lm14;

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