Volume 7, 1995

University of Bialystok

Copyright (c) 1995 Association of Mizar Users

### The abstract of the Mizar article:

### Definitions and Basic Properties of Boolean and Union of Many Sorted Sets

**by****Artur Kornilowicz**- Received April 27, 1995
- MML identifier: MBOOLEAN

- [ Mizar article, MML identifier index ]

environ vocabulary PBOOLE, FUNCT_1, ZF_REFLE, RELAT_1, FUNCT_4, CAT_1, BOOLE, CAT_4, FUNCOP_1, ZFMISC_1, PRALG_2, AUTALG_1, FUNCT_2, TARSKI, MATRIX_1, PRE_CIRC, FINSET_1; notation TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, FUNCT_1, FUNCT_2, FINSET_1, FUNCT_4, CQC_LANG, PBOOLE, PRALG_2, AUTALG_1, PRE_CIRC; constructors CQC_LANG, PRALG_2, AUTALG_1, PRE_CIRC, MEMBERED, XBOOLE_0; clusters SUBSET_1, PRE_CIRC, CQC_LANG, MEMBERED, ZFMISC_1, XBOOLE_0; requirements SUBSET, BOOLE; begin :: Boolean of Many Sorted Sets reserve x, y, I for set, A, B, X, Y for ManySortedSet of I; definition let I, A; func bool A -> ManySortedSet of I means :: MBOOLEAN:def 1 for i be set st i in I holds it.i = bool (A.i); end; definition let I, A; cluster bool A -> non-empty; end; theorem :: MBOOLEAN:1 :: Tarski:6 X = bool Y iff for A holds A in X iff A c= Y; theorem :: MBOOLEAN:2 :: ZFMISC_1:1 bool [0]I = I --> {{}}; theorem :: MBOOLEAN:3 bool (I --> x) = I --> bool x; theorem :: MBOOLEAN:4 :: ZFMISC_1:30 bool (I --> {x}) = I --> { {} , {x} }; theorem :: MBOOLEAN:5 :: ZFMISC_1:76 [0]I c= A; theorem :: MBOOLEAN:6 :: ZFMISC_1:79 A c= B implies bool A c= bool B; theorem :: MBOOLEAN:7 :: ZFMISC_1:81 bool A \/ bool B c= bool (A \/ B); theorem :: MBOOLEAN:8 :: ZFMISC_1:82 bool A \/ bool B = bool (A \/ B) implies for i be set st i in I holds A.i,B.i are_c=-comparable; theorem :: MBOOLEAN:9 :: ZFMISC_1:83 bool (A /\ B) = bool A /\ bool B; theorem :: MBOOLEAN:10 :: ZFMISC_1:84 bool (A \ B) c= (I --> {{}}) \/ (bool A \ bool B); theorem :: MBOOLEAN:11 :: ZFMISC_1:85 X c= A \ B iff X c= A & X misses B; theorem :: MBOOLEAN:12 :: ZFMISC_1:86 bool (A \ B) \/ bool (B \ A) c= bool (A \+\ B); theorem :: MBOOLEAN:13 :: ZFMISC_1:87 X c= A \+\ B iff X c= A \/ B & X misses A /\ B; canceled; theorem :: MBOOLEAN:15 :: ZFMISC_1:89 X c= A or Y c= A implies X /\ Y c= A; theorem :: MBOOLEAN:16 :: ZFMISC_1:90 X c= A implies X \ Y c= A; theorem :: MBOOLEAN:17 :: ZFMISC_1:91 X c= A & Y c= A implies X \+\ Y c= A; theorem :: MBOOLEAN:18 :: ZFMISC_1:105 [|X, Y|] c= bool bool (X \/ Y); theorem :: MBOOLEAN:19 :: FIN_TOPO:4 X c= A iff X in bool A; theorem :: MBOOLEAN:20 :: FRAENKEL:5 MSFuncs (A, B) c= bool [|A, B|]; begin :: Union of Many Sorted Sets definition let I, A; func union A -> ManySortedSet of I means :: MBOOLEAN:def 2 for i be set st i in I holds it.i = union (A.i); end; definition let I; cluster union [0]I -> empty-yielding; end; theorem :: MBOOLEAN:21 :: Tarski:def 4 A in union X iff ex Y st A in Y & Y in X; theorem :: MBOOLEAN:22 :: ZFMISC_1:2 union [0]I = [0]I; theorem :: MBOOLEAN:23 union (I --> x) = I --> union x; theorem :: MBOOLEAN:24 :: ZFMISC_1:31 union (I --> {x}) = I --> x; theorem :: MBOOLEAN:25 :: ZFMISC_1:32 union (I --> { {x},{y} }) = I --> {x,y}; theorem :: MBOOLEAN:26 :: ZFMISC_1:92 X in A implies X c= union A; theorem :: MBOOLEAN:27 :: ZFMISC_1:95 A c= B implies union A c= union B; theorem :: MBOOLEAN:28 :: ZFMISC_1:96 union (A \/ B) = union A \/ union B; theorem :: MBOOLEAN:29 :: ZFMISC_1:97 union (A /\ B) c= union A /\ union B; theorem :: MBOOLEAN:30 :: ZFMISC_1:99 union bool A = A; theorem :: MBOOLEAN:31 :: ZFMISC_1:100 A c= bool union A; theorem :: MBOOLEAN:32 :: LATTICE4:1 union Y c= A & X in Y implies X c= A; theorem :: MBOOLEAN:33 :: ZFMISC_1:94 for Z be ManySortedSet of I for A be non-empty ManySortedSet of I holds (for X be ManySortedSet of I st X in A holds X c= Z) implies union A c= Z; theorem :: MBOOLEAN:34 :: ZFMISC_1:98 for B be ManySortedSet of I for A be non-empty ManySortedSet of I holds (for X be ManySortedSet of I st X in A holds X /\ B = [0]I) implies union(A) /\ B = [0]I; theorem :: MBOOLEAN:35 :: ZFMISC_1:101 for A, B be ManySortedSet of I st A \/ B is non-empty holds (for X, Y be ManySortedSet of I st X <> Y & X in A \/ B & Y in A \/ B holds X /\ Y = [0]I) implies union(A /\ B) = union A /\ union B; theorem :: MBOOLEAN:36 :: LOPCLSET:31 for A, X be ManySortedSet of I for B be non-empty ManySortedSet of I holds (X c= union (A \/ B) & for Y be ManySortedSet of I st Y in B holds Y /\ X = [0]I) implies X c= union A; theorem :: MBOOLEAN:37 :: RLVECT_3:34 for A be locally-finite non-empty ManySortedSet of I st (for X, Y be ManySortedSet of I st X in A & Y in A holds X c= Y or Y c= X) holds union A in A;

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