Volume 11, 1999

University of Bialystok

Copyright (c) 1999 Association of Mizar Users

### The abstract of the Mizar article:

### Predicate Calculus for Boolean Valued Functions. Part III

**by****Shunichi Kobayashi, and****Yatsuka Nakamura**- Received July 14, 1999
- MML identifier: BVFUNC11

- [ Mizar article, MML identifier index ]

environ vocabulary EQREL_1, PARTIT1, T_1TOPSP, BOOLE, BVFUNC_2, FUNCOP_1, SETFAM_1, RELAT_1, FUNCT_1, CANTOR_1, CAT_1, FUNCT_2, MARGREL1, ZF_LANG, BVFUNC_1; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, MARGREL1, VALUAT_1, RELAT_1, SETFAM_1, FUNCT_1, FRAENKEL, EQREL_1, CANTOR_1, CQC_LANG, PARTIT1, BVFUNC_1, BVFUNC_2, FUNCT_4; constructors CANTOR_1, BVFUNC_2, PUA2MSS1, BVFUNC_1, FUNCT_4, MEMBERED; clusters FUNCT_7, PARTIT1, SUBSET_1, MARGREL1, VALUAT_1, FRAENKEL; requirements SUBSET, BOOLE; begin ::Chap. 1 Preliminaries reserve Y for non empty set; theorem :: BVFUNC11:1 for z being Element of Y, PA,PB being a_partition of Y st PA '<' PB holds EqClass(z,PA) c= EqClass(z,PB); theorem :: BVFUNC11:2 for z being Element of Y, PA,PB being a_partition of Y holds EqClass(z,PA) c= EqClass(z,PA '\/' PB); theorem :: BVFUNC11:3 for z being Element of Y, PA,PB being a_partition of Y holds EqClass(z,PA '/\' PB) c= EqClass(z,PA); theorem :: BVFUNC11:4 for z being Element of Y, PA being a_partition of Y holds EqClass(z,PA) c= EqClass(z,%O(Y)) & EqClass(z,%I(Y)) c= EqClass(z,PA); theorem :: BVFUNC11:5 for G being Subset of PARTITIONS(Y), A,B being a_partition of Y st G is independent & G={A,B} & A<>B holds for a,b being set st a in A & b in B holds a meets b; theorem :: BVFUNC11:6 for G being Subset of PARTITIONS(Y), A,B being a_partition of Y st G is independent & G={A,B} & A <> B holds '/\' G = A '/\' B; theorem :: BVFUNC11:7 for G being Subset of PARTITIONS(Y), A,B being a_partition of Y st G={A,B} & A<>B holds CompF(A,G) = B; begin canceled 3; theorem :: BVFUNC11:11 for a being Element of Funcs(Y,BOOLEAN), G being Subset of PARTITIONS(Y), A,B being a_partition of Y st G is independent holds All(All(a,A,G),B,G) '<' Ex(All(a,B,G),A,G); theorem :: BVFUNC11:12 for a being Element of Funcs(Y,BOOLEAN), G being Subset of PARTITIONS(Y), A,B being a_partition of Y holds All(All(a,A,G),B,G) '<' Ex(Ex(a,B,G),A,G); theorem :: BVFUNC11:13 for a being Element of Funcs(Y,BOOLEAN), G being Subset of PARTITIONS(Y), A,B being a_partition of Y st G is independent holds All(All(a,A,G),B,G) '<' All(Ex(a,B,G),A,G); theorem :: BVFUNC11:14 for a being Element of Funcs(Y,BOOLEAN),G being Subset of PARTITIONS(Y), A,B being a_partition of Y holds All(Ex(a,A,G),B,G) '<' Ex(Ex(a,B,G),A,G); theorem :: BVFUNC11:15 for a being Element of Funcs(Y,BOOLEAN), G being Subset of PARTITIONS(Y), A,B being a_partition of Y holds 'not' Ex(All(a,A,G),B,G) '<' Ex(Ex('not' a,B,G),A,G); theorem :: BVFUNC11:16 for a being Element of Funcs(Y,BOOLEAN), G being Subset of PARTITIONS(Y), A,B being a_partition of Y st G is independent holds Ex('not' All(a,A,G),B,G) '<' Ex(Ex('not' a,B,G),A,G); theorem :: BVFUNC11:17 for a being Element of Funcs(Y,BOOLEAN), G being Subset of PARTITIONS(Y), A,B being a_partition of Y st G is independent holds 'not' All(All(a,A,G),B,G) = Ex('not' All(a,B,G),A,G); theorem :: BVFUNC11:18 for a being Element of Funcs(Y,BOOLEAN), G being Subset of PARTITIONS(Y), A,B being a_partition of Y st G is independent holds All('not' All(a,A,G),B,G) '<' Ex(Ex('not' a,B,G),A,G); theorem :: BVFUNC11:19 for a being Element of Funcs(Y,BOOLEAN), G being Subset of PARTITIONS(Y), A,B being a_partition of Y st G is independent holds 'not' All(All(a,A,G),B,G) = Ex(Ex('not' a,B,G),A,G); theorem :: BVFUNC11:20 for a being Element of Funcs(Y,BOOLEAN), G being Subset of PARTITIONS(Y), A,B being a_partition of Y st G is independent holds 'not' All(All(a,A,G),B,G) '<' Ex(Ex('not' a,A,G),B,G);

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