Volume 1, 1989

University of Bialystok

Copyright (c) 1989 Association of Mizar Users

### The abstract of the Mizar article:

### Properties of ZF Models

**by****Grzegorz Bancerek**- Received July 5, 1989
- MML identifier: ZFMODEL1

- [ Mizar article, MML identifier index ]

environ vocabulary ZF_LANG, FUNCT_1, ORDINAL1, ZF_MODEL, TARSKI, BOOLE, FINSEQ_1, MCART_1, RELAT_1, ZFMODEL1; notation TARSKI, XBOOLE_0, ENUMSET1, ZFMISC_1, SUBSET_1, XREAL_0, NAT_1, RELAT_1, FUNCT_1, FINSEQ_1, ZF_LANG, RELSET_1, FUNCT_2, ORDINAL1, ZF_MODEL, MCART_1; constructors NAT_1, ENUMSET1, ZF_MODEL, MCART_1, XREAL_0, MEMBERED, XBOOLE_0; clusters ZF_LANG, RELSET_1, SUBSET_1, MEMBERED, ZFMISC_1, XBOOLE_0; requirements NUMERALS, SUBSET, BOOLE, ARITHM; begin reserve x,y,z for Variable, H for ZF-formula, E for non empty set, a,b,c,X,Y,Z for set, u,v,w for Element of E, f,g,h,i,j for Function of VAR,E, k,n for Nat; theorem :: ZFMODEL1:1 E is epsilon-transitive implies E |= the_axiom_of_extensionality; theorem :: ZFMODEL1:2 E is epsilon-transitive implies (E |= the_axiom_of_pairs iff for u,v holds { u,v } in E); theorem :: ZFMODEL1:3 E is epsilon-transitive implies (E |= the_axiom_of_pairs iff for X,Y st X in E & Y in E holds { X,Y } in E); theorem :: ZFMODEL1:4 E is epsilon-transitive implies (E |= the_axiom_of_unions iff for u holds union u in E); theorem :: ZFMODEL1:5 E is epsilon-transitive implies (E |= the_axiom_of_unions iff for X st X in E holds union X in E); theorem :: ZFMODEL1:6 E is epsilon-transitive implies (E |= the_axiom_of_infinity iff ex u st u <> {} & for v st v in u ex w st v c< w & w in u); theorem :: ZFMODEL1:7 E is epsilon-transitive implies (E |= the_axiom_of_infinity iff ex X st X in E & X <> {} & for Y st Y in X ex Z st Y c< Z & Z in X); theorem :: ZFMODEL1:8 E is epsilon-transitive implies (E |= the_axiom_of_power_sets iff for u holds E /\ bool u in E); theorem :: ZFMODEL1:9 E is epsilon-transitive implies (E |= the_axiom_of_power_sets iff for X st X in E holds E /\ bool X in E); theorem :: ZFMODEL1:10 not x in Free H & E,f |= H implies E,f |= All(x,H); theorem :: ZFMODEL1:11 { x,y } misses Free H & E,f |= H implies E,f |= All(x,y,H); theorem :: ZFMODEL1:12 { x,y,z } misses Free H & E,f |= H implies E,f |= All(x,y,z,H); definition let H,E; let val be Function of VAR,E; assume not x.0 in Free H & E,val |= All(x.3,Ex(x.0,All(x.4,H <=> x.4 '=' x.0))); func def_func'(H,val) -> Function of E,E means :: ZFMODEL1:def 1 for g st for y st g.y <> val.y holds x.0 = y or x.3 = y or x.4 = y holds E,g |= H iff it.(g.x.3) = g.x.4; end; canceled; theorem :: ZFMODEL1:14 for H,f,g st (for x st f.x <> g.x holds not x in Free H) & E,f |= H holds E,g |= H; definition let H,E; assume Free H c= { x.3,x.4 } & E |= All(x.3,Ex(x.0,All(x.4,H <=> x.4 '=' x.0))); func def_func(H,E) -> Function of E,E means :: ZFMODEL1:def 2 for g holds E,g |= H iff it.(g.x.3) = g.x.4; end; definition let F be Function; let E; pred F is_definable_in E means :: ZFMODEL1:def 3 ex H st Free H c= { x.3,x.4 } & E |= All(x.3,Ex(x.0,All(x.4,H <=> x.4 '=' x.0))) & F = def_func(H,E); pred F is_parametrically_definable_in E means :: ZFMODEL1:def 4 ex H,f st { x.0,x.1,x.2 } misses Free H & E,f |= All(x.3,Ex(x.0,All(x.4,H <=> x.4 '=' x.0))) & F = def_func'(H,f); end; canceled 3; theorem :: ZFMODEL1:18 for F being Function st F is_definable_in E holds F is_parametrically_definable_in E; theorem :: ZFMODEL1:19 E is epsilon-transitive implies ((for H st { x.0,x.1,x.2 } misses Free H holds E |= the_axiom_of_substitution_for H) iff for H,f st { x.0,x.1,x.2 } misses Free H & E,f |= All(x.3,Ex(x.0,All(x.4,H <=> x.4 '=' x.0))) for u holds def_func'(H,f).:u in E ); theorem :: ZFMODEL1:20 E is epsilon-transitive implies ((for H st { x.0,x.1,x.2 } misses Free H holds E |= the_axiom_of_substitution_for H) iff for F being Function st F is_parametrically_definable_in E for X st X in E holds F.:X in E ); theorem :: ZFMODEL1:21 E is_a_model_of_ZF implies E is epsilon-transitive & (for u,v st for w holds w in u iff w in v holds u = v) & (for u,v holds { u,v } in E) & (for u holds union u in E) & (ex u st u <> {} & for v st v in u ex w st v c< w & w in u) & (for u holds E /\ bool u in E) & (for H,f st { x.0,x.1,x.2 } misses Free H & E,f |= All(x.3,Ex(x.0,All(x.4,H <=> x.4 '=' x.0))) for u holds def_func'(H,f).:u in E ); theorem :: ZFMODEL1:22 E is epsilon-transitive & (for u,v holds { u,v } in E) & (for u holds union u in E) & (ex u st u <> {} & for v st v in u ex w st v c< w & w in u) & (for u holds E /\ bool u in E) & (for H,f st { x.0,x.1,x.2 } misses Free H & E,f |= All(x.3,Ex(x.0,All(x.4,H <=> x.4 '=' x.0))) for u holds def_func'(H,f).:u in E ) implies E is_a_model_of_ZF;

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