Volume 2, 1990

University of Bialystok

Copyright (c) 1990 Association of Mizar Users

### The abstract of the Mizar article:

### Tarski's Classes and Ranks

**by****Grzegorz Bancerek**- Received March 23, 1990
- MML identifier: CLASSES1

- [ Mizar article, MML identifier index ]

environ vocabulary FUNCT_1, TARSKI, SETFAM_1, BOOLE, CARD_1, ORDINAL1, ORDINAL2, RELAT_1, MCART_1, CLASSES1; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, SETFAM_1, RELAT_1, FUNCT_1, NUMBERS, NAT_1, ORDINAL1, MCART_1, ORDINAL2, CARD_1; constructors SETFAM_1, NAT_1, MCART_1, WELLORD2, CARD_1, MEMBERED, XBOOLE_0; clusters SUBSET_1, ORDINAL1, MEMBERED, ZFMISC_1, XBOOLE_0, NUMBERS, ORDINAL2; requirements NUMERALS, SUBSET, BOOLE; begin reserve W,X,Y,Z for set, f,g for Function, a,x,y,z for set; definition let B be set; attr B is subset-closed means :: CLASSES1:def 1 for X,Y holds X in B & Y c= X implies Y in B; end; definition let B be set; attr B is being_Tarski-Class means :: CLASSES1:def 2 B is subset-closed & (for X holds X in B implies bool X in B) & (for X holds X c= B implies X,B are_equipotent or X in B); synonym B is_Tarski-Class; end; definition let A,B be set; pred B is_Tarski-Class_of A means :: CLASSES1:def 3 A in B & B is_Tarski-Class; end; definition let A be set; func Tarski-Class A -> set means :: CLASSES1:def 4 it is_Tarski-Class_of A & for D being set st D is_Tarski-Class_of A holds it c= D; end; definition let A be set; cluster Tarski-Class A -> non empty; end; canceled; theorem :: CLASSES1:2 W is_Tarski-Class iff W is subset-closed & (for X st X in W holds bool X in W) & (for X st X c= W & Card X <` Card W holds X in W); canceled 2; theorem :: CLASSES1:5 X in Tarski-Class X; theorem :: CLASSES1:6 Y in Tarski-Class X & Z c= Y implies Z in Tarski-Class X; theorem :: CLASSES1:7 Y in Tarski-Class X implies bool Y in Tarski-Class X; theorem :: CLASSES1:8 Y c= Tarski-Class X implies Y,Tarski-Class X are_equipotent or Y in Tarski-Class X; theorem :: CLASSES1:9 Y c= Tarski-Class X & Card Y <` Card Tarski-Class X implies Y in Tarski-Class X; reserve u,v for Element of Tarski-Class(X), A,B,C for Ordinal, L,L1 for T-Sequence; definition let X,A; func Tarski-Class(X,A) means :: CLASSES1:def 5 ex L st it = last L & dom L = succ A & L.{} = { X } & (for C st succ C in succ A holds L.succ C = { u : ex v st v in L.C & u c= v } \/ { bool v : v in L.C } \/ bool(L.C) /\ Tarski-Class X) & for C st C in succ A & C <> {} & C is_limit_ordinal holds L.C = (union rng(L|C)) /\ Tarski-Class X; end; definition let X,A; redefine func Tarski-Class(X,A) -> Subset of Tarski-Class X; end; theorem :: CLASSES1:10 Tarski-Class(X,{}) = { X }; theorem :: CLASSES1:11 Tarski-Class(X,succ A) = { u : ex v st v in Tarski-Class(X,A) & u c= v } \/ { bool v : v in Tarski-Class(X,A) } \/ bool Tarski-Class(X,A) /\ Tarski-Class X; theorem :: CLASSES1:12 A <> {} & A is_limit_ordinal implies Tarski-Class(X,A) = { u : ex B st B in A & u in Tarski-Class(X,B) }; theorem :: CLASSES1:13 Y in Tarski-Class(X,succ A) iff Y c= Tarski-Class(X,A) & Y in Tarski-Class X or ex Z st Z in Tarski-Class(X,A) & (Y c= Z or Y = bool Z); theorem :: CLASSES1:14 Y c= Z & Z in Tarski-Class(X,A) implies Y in Tarski-Class(X,succ A); theorem :: CLASSES1:15 Y in Tarski-Class(X,A) implies bool Y in Tarski-Class(X,succ A); theorem :: CLASSES1:16 A <> {} & A is_limit_ordinal implies (x in Tarski-Class(X,A) iff ex B st B in A & x in Tarski-Class(X,B)); theorem :: CLASSES1:17 A <> {} & A is_limit_ordinal & Y in Tarski-Class(X,A) & (Z c= Y or Z = bool Y) implies Z in Tarski-Class(X,A); theorem :: CLASSES1:18 Tarski-Class(X,A) c= Tarski-Class(X,succ A); theorem :: CLASSES1:19 A c= B implies Tarski-Class(X,A) c= Tarski-Class(X,B); theorem :: CLASSES1:20 ex A st Tarski-Class(X,A) = Tarski-Class(X,succ A); theorem :: CLASSES1:21 Tarski-Class(X,A) = Tarski-Class(X,succ A) implies Tarski-Class(X,A) = Tarski-Class X; theorem :: CLASSES1:22 ex A st Tarski-Class(X,A) = Tarski-Class X; theorem :: CLASSES1:23 ex A st Tarski-Class(X,A) = Tarski-Class X & for B st B in A holds Tarski-Class(X,B) <> Tarski-Class X; theorem :: CLASSES1:24 Y <> X & Y in Tarski-Class X implies ex A st not Y in Tarski-Class(X,A) & Y in Tarski-Class(X,succ A); theorem :: CLASSES1:25 X is epsilon-transitive implies for A st A <> {} holds Tarski-Class(X,A) is epsilon-transitive; theorem :: CLASSES1:26 Tarski-Class(X,{}) in Tarski-Class(X,one) & Tarski-Class(X,{}) <> Tarski-Class(X,one); theorem :: CLASSES1:27 X is epsilon-transitive implies Tarski-Class X is epsilon-transitive; theorem :: CLASSES1:28 Y in Tarski-Class X implies Card Y <` Card Tarski-Class X; theorem :: CLASSES1:29 Y in Tarski-Class X implies not Y,Tarski-Class X are_equipotent; theorem :: CLASSES1:30 x in Tarski-Class X & y in Tarski-Class X implies {x} in Tarski-Class X & {x,y} in Tarski-Class X; theorem :: CLASSES1:31 x in Tarski-Class X & y in Tarski-Class X implies [x,y] in Tarski-Class X; theorem :: CLASSES1:32 Y c= Tarski-Class X & Z c= Tarski-Class X implies [:Y,Z:] c= Tarski-Class X; definition let A; func Rank(A) means :: CLASSES1:def 6 ex L st it = last L & dom L = succ A & L.{} = {} & (for C st succ C in succ A holds L.succ C = bool(L.C)) & for C st C in succ A & C <> {} & C is_limit_ordinal holds L.C = union rng(L|C); end; theorem :: CLASSES1:33 Rank {} = {}; theorem :: CLASSES1:34 Rank succ A = bool Rank A; theorem :: CLASSES1:35 A <> {} & A is_limit_ordinal implies for x holds x in Rank A iff ex B st B in A & x in Rank B; theorem :: CLASSES1:36 X c= Rank A iff X in Rank succ A; theorem :: CLASSES1:37 Rank A is epsilon-transitive; theorem :: CLASSES1:38 X in Rank A implies X c= Rank A; theorem :: CLASSES1:39 Rank A c= Rank succ A; theorem :: CLASSES1:40 union Rank A c= Rank A; theorem :: CLASSES1:41 X in Rank A implies union X in Rank A; theorem :: CLASSES1:42 A in B iff Rank A in Rank B; theorem :: CLASSES1:43 A c= B iff Rank A c= Rank B; theorem :: CLASSES1:44 A c= Rank A; theorem :: CLASSES1:45 for A,X st X in Rank A holds not X,Rank A are_equipotent & Card X <` Card Rank A; theorem :: CLASSES1:46 X c= Rank A iff bool X c= Rank succ A; theorem :: CLASSES1:47 X c= Y & Y in Rank A implies X in Rank A; theorem :: CLASSES1:48 X in Rank A iff bool X in Rank succ A; theorem :: CLASSES1:49 x in Rank A iff {x} in Rank succ A; theorem :: CLASSES1:50 x in Rank A & y in Rank A iff {x,y} in Rank succ A; theorem :: CLASSES1:51 x in Rank A & y in Rank A iff [x,y] in Rank succ succ A; theorem :: CLASSES1:52 X is epsilon-transitive & Rank A /\ Tarski-Class X = Rank succ A /\ Tarski-Class X implies Tarski-Class X c= Rank A; theorem :: CLASSES1:53 X is epsilon-transitive implies ex A st Tarski-Class X c= Rank A; theorem :: CLASSES1:54 X is epsilon-transitive implies union X c= X; theorem :: CLASSES1:55 X is epsilon-transitive & Y is epsilon-transitive implies X \/ Y is epsilon-transitive; theorem :: CLASSES1:56 X is epsilon-transitive & Y is epsilon-transitive implies X /\ Y is epsilon-transitive; reserve k,n for Nat; definition let X; func the_transitive-closure_of X -> set means :: CLASSES1:def 7 x in it iff ex f,n st x in f.n & dom f = NAT & f.0 = X & for k holds f.(k+1) = union(f.k); end; canceled; theorem :: CLASSES1:58 the_transitive-closure_of X is epsilon-transitive; theorem :: CLASSES1:59 X c= the_transitive-closure_of X; theorem :: CLASSES1:60 X c= Y & Y is epsilon-transitive implies the_transitive-closure_of X c= Y; theorem :: CLASSES1:61 (for Z st X c= Z & Z is epsilon-transitive holds Y c= Z) & X c= Y & Y is epsilon-transitive implies the_transitive-closure_of X = Y; theorem :: CLASSES1:62 X is epsilon-transitive implies the_transitive-closure_of X = X; theorem :: CLASSES1:63 the_transitive-closure_of {} = {}; theorem :: CLASSES1:64 the_transitive-closure_of A = A; theorem :: CLASSES1:65 X c= Y implies the_transitive-closure_of X c= the_transitive-closure_of Y; theorem :: CLASSES1:66 the_transitive-closure_of the_transitive-closure_of X = the_transitive-closure_of X; theorem :: CLASSES1:67 the_transitive-closure_of (X \/ Y) = the_transitive-closure_of X \/ the_transitive-closure_of Y; theorem :: CLASSES1:68 the_transitive-closure_of (X /\ Y) c= the_transitive-closure_of X /\ the_transitive-closure_of Y; theorem :: CLASSES1:69 ex A st X c= Rank A; definition let X; func the_rank_of X -> Ordinal means :: CLASSES1:def 8 X c= Rank it & for B st X c= Rank B holds it c= B; end; canceled; theorem :: CLASSES1:71 the_rank_of bool X = succ the_rank_of X; theorem :: CLASSES1:72 the_rank_of Rank A = A; theorem :: CLASSES1:73 X c= Rank A iff the_rank_of X c= A; theorem :: CLASSES1:74 X in Rank A iff the_rank_of X in A; theorem :: CLASSES1:75 X c= Y implies the_rank_of X c= the_rank_of Y; theorem :: CLASSES1:76 X in Y implies the_rank_of X in the_rank_of Y; theorem :: CLASSES1:77 the_rank_of X c= A iff for Y st Y in X holds the_rank_of Y in A; theorem :: CLASSES1:78 A c= the_rank_of X iff for B st B in A ex Y st Y in X & B c= the_rank_of Y; theorem :: CLASSES1:79 the_rank_of X = {} iff X = {}; theorem :: CLASSES1:80 the_rank_of X = succ A implies ex Y st Y in X & the_rank_of Y = A; theorem :: CLASSES1:81 the_rank_of A = A; theorem :: CLASSES1:82 the_rank_of Tarski-Class X <> {} & the_rank_of Tarski-Class X is_limit_ordinal;

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