SET007 Axioms: SET007+72.ax
%------------------------------------------------------------------------------
% File : SET007+72 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Properties of ZF Models
% Version : [Urb08] axioms.
% English :
% Refs : [Mat90] Matuszewski (1990), Formalized Mathematics
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : zfmodel1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 28 ( 4 unt; 0 def)
% Number of atoms : 265 ( 16 equ)
% Maximal formula atoms : 32 ( 9 avg)
% Number of connectives : 281 ( 44 ~; 3 |; 100 &)
% ( 17 <=>; 117 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 10 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of predicates : 23 ( 21 usr; 1 prp; 0-3 aty)
% Number of functors : 32 ( 32 usr; 13 con; 0-4 aty)
% Number of variables : 101 ( 95 !; 6 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(t1_zfmodel1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_ordinal1(A)
=> r2_zf_model(A,k6_zf_model) ) ) ).
fof(t2_zfmodel1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_ordinal1(A)
=> ( r2_zf_model(A,k7_zf_model)
<=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> r2_hidden(k2_tarski(B,C),A) ) ) ) ) ) ).
fof(t3_zfmodel1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_ordinal1(A)
=> ( r2_zf_model(A,k7_zf_model)
<=> ! [B,C] :
( ( r2_hidden(B,A)
& r2_hidden(C,A) )
=> r2_hidden(k2_tarski(B,C),A) ) ) ) ) ).
fof(t4_zfmodel1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_ordinal1(A)
=> ( r2_zf_model(A,k8_zf_model)
<=> ! [B] :
( m1_subset_1(B,A)
=> r2_hidden(k3_tarski(B),A) ) ) ) ) ).
fof(t5_zfmodel1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_ordinal1(A)
=> ( r2_zf_model(A,k8_zf_model)
<=> ! [B] :
( r2_hidden(B,A)
=> r2_hidden(k3_tarski(B),A) ) ) ) ) ).
fof(t6_zfmodel1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_ordinal1(A)
=> ( r2_zf_model(A,k9_zf_model)
<=> ? [B] :
( m1_subset_1(B,A)
& B != k1_xboole_0
& ! [C] :
( m1_subset_1(C,A)
=> ~ ( r2_hidden(C,B)
& ! [D] :
( m1_subset_1(D,A)
=> ~ ( r2_xboole_0(C,D)
& r2_hidden(D,B) ) ) ) ) ) ) ) ) ).
fof(t7_zfmodel1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_ordinal1(A)
=> ( r2_zf_model(A,k9_zf_model)
<=> ? [B] :
( r2_hidden(B,A)
& B != k1_xboole_0
& ! [C] :
~ ( r2_hidden(C,B)
& ! [D] :
~ ( r2_xboole_0(C,D)
& r2_hidden(D,B) ) ) ) ) ) ) ).
fof(t8_zfmodel1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_ordinal1(A)
=> ( r2_zf_model(A,k10_zf_model)
<=> ! [B] :
( m1_subset_1(B,A)
=> r2_hidden(k3_xboole_0(A,k1_zfmisc_1(B)),A) ) ) ) ) ).
fof(t9_zfmodel1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_ordinal1(A)
=> ( r2_zf_model(A,k10_zf_model)
<=> ! [B] :
( r2_hidden(B,A)
=> r2_hidden(k3_xboole_0(A,k1_zfmisc_1(B)),A) ) ) ) ) ).
fof(t10_zfmodel1,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k1_zf_lang,C)
& m2_relset_1(D,k1_zf_lang,C) )
=> ( r1_zf_model(C,D,B)
=> ( r2_hidden(A,k2_zf_model(B))
| r1_zf_model(C,D,k8_zf_lang(A,B)) ) ) ) ) ) ) ).
fof(t11_zfmodel1,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ! [D] :
( ~ v1_xboole_0(D)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k1_zf_lang,D)
& m2_relset_1(E,k1_zf_lang,D) )
=> ( ( r1_xboole_0(k2_tarski(A,B),k2_zf_model(C))
& r1_zf_model(D,E,C) )
=> r1_zf_model(D,E,k14_zf_lang(A,B,C)) ) ) ) ) ) ) ).
fof(t12_zfmodel1,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ! [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
=> ! [D] :
( ( v1_zf_lang(D)
& m2_finseq_1(D,k5_numbers) )
=> ! [E] :
( ~ v1_xboole_0(E)
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k1_zf_lang,E)
& m2_relset_1(F,k1_zf_lang,E) )
=> ( ( r1_xboole_0(k1_enumset1(A,B,C),k2_zf_model(D))
& r1_zf_model(E,F,D) )
=> r1_zf_model(E,F,k16_zf_lang(A,B,C,D)) ) ) ) ) ) ) ) ).
fof(d1_zfmodel1,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_zf_lang,B)
& m2_relset_1(C,k1_zf_lang,B) )
=> ( r1_zf_model(B,C,k8_zf_lang(k2_zf_lang(np__3),k13_zf_lang(k2_zf_lang(np__0),k8_zf_lang(k2_zf_lang(np__4),k12_zf_lang(A,k4_zf_lang(k2_zf_lang(np__4),k2_zf_lang(np__0)))))))
=> ( r2_hidden(k2_zf_lang(np__0),k2_zf_model(A))
| ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,B,B)
& m2_relset_1(D,B,B) )
=> ( D = k1_zfmodel1(A,B,C)
<=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k1_zf_lang,B)
& m2_relset_1(E,k1_zf_lang,B) )
=> ( ! [F] :
( m2_subset_1(F,k5_numbers,k1_zf_lang)
=> ~ ( k8_funct_2(k1_zf_lang,B,E,F) != k8_funct_2(k1_zf_lang,B,C,F)
& k2_zf_lang(np__0) != F
& k2_zf_lang(np__3) != F
& k2_zf_lang(np__4) != F ) )
=> ( r1_zf_model(B,E,A)
<=> k8_funct_2(B,B,D,k8_funct_2(k1_zf_lang,B,E,k2_zf_lang(np__3))) = k8_funct_2(k1_zf_lang,B,E,k2_zf_lang(np__4)) ) ) ) ) ) ) ) ) ) ) ).
fof(t13_zfmodel1,axiom,
$true ).
fof(t14_zfmodel1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_zf_lang,A)
& m2_relset_1(C,k1_zf_lang,A) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k1_zf_lang,A)
& m2_relset_1(D,k1_zf_lang,A) )
=> ( ( ! [E] :
( m2_subset_1(E,k5_numbers,k1_zf_lang)
=> ~ ( k8_funct_2(k1_zf_lang,A,C,E) != k8_funct_2(k1_zf_lang,A,D,E)
& r2_hidden(E,k2_zf_model(B)) ) )
& r1_zf_model(A,C,B) )
=> r1_zf_model(A,D,B) ) ) ) ) ) ).
fof(d2_zfmodel1,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ~ v1_xboole_0(B)
=> ( ( r1_tarski(k2_zf_model(A),k2_tarski(k2_zf_lang(np__3),k2_zf_lang(np__4)))
& r2_zf_model(B,k8_zf_lang(k2_zf_lang(np__3),k13_zf_lang(k2_zf_lang(np__0),k8_zf_lang(k2_zf_lang(np__4),k12_zf_lang(A,k4_zf_lang(k2_zf_lang(np__4),k2_zf_lang(np__0))))))) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,B)
& m2_relset_1(C,B,B) )
=> ( C = k2_zfmodel1(A,B)
<=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k1_zf_lang,B)
& m2_relset_1(D,k1_zf_lang,B) )
=> ( r1_zf_model(B,D,A)
<=> k8_funct_2(B,B,C,k8_funct_2(k1_zf_lang,B,D,k2_zf_lang(np__3))) = k8_funct_2(k1_zf_lang,B,D,k2_zf_lang(np__4)) ) ) ) ) ) ) ) ).
fof(d3_zfmodel1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ~ v1_xboole_0(B)
=> ( r1_zfmodel1(A,B)
<=> ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& r1_tarski(k2_zf_model(C),k2_tarski(k2_zf_lang(np__3),k2_zf_lang(np__4)))
& r2_zf_model(B,k8_zf_lang(k2_zf_lang(np__3),k13_zf_lang(k2_zf_lang(np__0),k8_zf_lang(k2_zf_lang(np__4),k12_zf_lang(C,k4_zf_lang(k2_zf_lang(np__4),k2_zf_lang(np__0)))))))
& A = k2_zfmodel1(C,B) ) ) ) ) ).
fof(d4_zfmodel1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ~ v1_xboole_0(B)
=> ( r2_zfmodel1(A,B)
<=> ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& ? [D] :
( v1_funct_1(D)
& v1_funct_2(D,k1_zf_lang,B)
& m2_relset_1(D,k1_zf_lang,B)
& r1_xboole_0(k1_enumset1(k2_zf_lang(np__0),k2_zf_lang(np__1),k2_zf_lang(np__2)),k2_zf_model(C))
& r1_zf_model(B,D,k8_zf_lang(k2_zf_lang(np__3),k13_zf_lang(k2_zf_lang(np__0),k8_zf_lang(k2_zf_lang(np__4),k12_zf_lang(C,k4_zf_lang(k2_zf_lang(np__4),k2_zf_lang(np__0)))))))
& A = k1_zfmodel1(C,B,D) ) ) ) ) ) ).
fof(t15_zfmodel1,axiom,
$true ).
fof(t16_zfmodel1,axiom,
$true ).
fof(t17_zfmodel1,axiom,
$true ).
fof(t18_zfmodel1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_zfmodel1(B,A)
=> r2_zfmodel1(B,A) ) ) ) ).
fof(t19_zfmodel1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_ordinal1(A)
=> ( ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ( r1_xboole_0(k1_enumset1(k2_zf_lang(np__0),k2_zf_lang(np__1),k2_zf_lang(np__2)),k2_zf_model(B))
=> r2_zf_model(A,k11_zf_model(B)) ) )
<=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_zf_lang,A)
& m2_relset_1(C,k1_zf_lang,A) )
=> ( ( r1_xboole_0(k1_enumset1(k2_zf_lang(np__0),k2_zf_lang(np__1),k2_zf_lang(np__2)),k2_zf_model(B))
& r1_zf_model(A,C,k8_zf_lang(k2_zf_lang(np__3),k13_zf_lang(k2_zf_lang(np__0),k8_zf_lang(k2_zf_lang(np__4),k12_zf_lang(B,k4_zf_lang(k2_zf_lang(np__4),k2_zf_lang(np__0))))))) )
=> ! [D] :
( m1_subset_1(D,A)
=> r2_hidden(k2_funct_2(A,A,k1_zfmodel1(B,A,C),D),A) ) ) ) ) ) ) ) ).
fof(t20_zfmodel1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_ordinal1(A)
=> ( ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ( r1_xboole_0(k1_enumset1(k2_zf_lang(np__0),k2_zf_lang(np__1),k2_zf_lang(np__2)),k2_zf_model(B))
=> r2_zf_model(A,k11_zf_model(B)) ) )
<=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r2_zfmodel1(B,A)
=> ! [C] :
( r2_hidden(C,A)
=> r2_hidden(k9_relat_1(B,C),A) ) ) ) ) ) ) ).
fof(t21_zfmodel1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_zf_model(A)
=> ( v1_ordinal1(A)
& ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ( ! [D] :
( m1_subset_1(D,A)
=> ( r2_hidden(D,B)
<=> r2_hidden(D,C) ) )
=> B = C ) ) )
& ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> r2_hidden(k2_tarski(B,C),A) ) )
& ! [B] :
( m1_subset_1(B,A)
=> r2_hidden(k3_tarski(B),A) )
& ? [B] :
( m1_subset_1(B,A)
& B != k1_xboole_0
& ! [C] :
( m1_subset_1(C,A)
=> ~ ( r2_hidden(C,B)
& ! [D] :
( m1_subset_1(D,A)
=> ~ ( r2_xboole_0(C,D)
& r2_hidden(D,B) ) ) ) ) )
& ! [B] :
( m1_subset_1(B,A)
=> r2_hidden(k3_xboole_0(A,k1_zfmisc_1(B)),A) )
& ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_zf_lang,A)
& m2_relset_1(C,k1_zf_lang,A) )
=> ( ( r1_xboole_0(k1_enumset1(k2_zf_lang(np__0),k2_zf_lang(np__1),k2_zf_lang(np__2)),k2_zf_model(B))
& r1_zf_model(A,C,k8_zf_lang(k2_zf_lang(np__3),k13_zf_lang(k2_zf_lang(np__0),k8_zf_lang(k2_zf_lang(np__4),k12_zf_lang(B,k4_zf_lang(k2_zf_lang(np__4),k2_zf_lang(np__0))))))) )
=> ! [D] :
( m1_subset_1(D,A)
=> r2_hidden(k2_funct_2(A,A,k1_zfmodel1(B,A,C),D),A) ) ) ) ) ) ) ) ).
fof(t22_zfmodel1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ( v1_ordinal1(A)
& ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> r2_hidden(k2_tarski(B,C),A) ) )
& ! [B] :
( m1_subset_1(B,A)
=> r2_hidden(k3_tarski(B),A) )
& ! [B] :
( m1_subset_1(B,A)
=> r2_hidden(k3_xboole_0(A,k1_zfmisc_1(B)),A) )
& ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_zf_lang,A)
& m2_relset_1(C,k1_zf_lang,A) )
=> ( ( r1_xboole_0(k1_enumset1(k2_zf_lang(np__0),k2_zf_lang(np__1),k2_zf_lang(np__2)),k2_zf_model(B))
& r1_zf_model(A,C,k8_zf_lang(k2_zf_lang(np__3),k13_zf_lang(k2_zf_lang(np__0),k8_zf_lang(k2_zf_lang(np__4),k12_zf_lang(B,k4_zf_lang(k2_zf_lang(np__4),k2_zf_lang(np__0))))))) )
=> ! [D] :
( m1_subset_1(D,A)
=> r2_hidden(k2_funct_2(A,A,k1_zfmodel1(B,A,C),D),A) ) ) ) ) )
=> ( ! [B] :
( m1_subset_1(B,A)
=> ~ ( B != k1_xboole_0
& ! [C] :
( m1_subset_1(C,A)
=> ~ ( r2_hidden(C,B)
& ! [D] :
( m1_subset_1(D,A)
=> ~ ( r2_xboole_0(C,D)
& r2_hidden(D,B) ) ) ) ) ) )
| v1_zf_model(A) ) ) ) ).
fof(dt_k1_zfmodel1,axiom,
! [A,B,C] :
( ( v1_zf_lang(A)
& m1_finseq_1(A,k5_numbers)
& ~ v1_xboole_0(B)
& v1_funct_1(C)
& v1_funct_2(C,k1_zf_lang,B)
& m1_relset_1(C,k1_zf_lang,B) )
=> ( v1_funct_1(k1_zfmodel1(A,B,C))
& v1_funct_2(k1_zfmodel1(A,B,C),B,B)
& m2_relset_1(k1_zfmodel1(A,B,C),B,B) ) ) ).
fof(dt_k2_zfmodel1,axiom,
! [A,B] :
( ( v1_zf_lang(A)
& m1_finseq_1(A,k5_numbers)
& ~ v1_xboole_0(B) )
=> ( v1_funct_1(k2_zfmodel1(A,B))
& v1_funct_2(k2_zfmodel1(A,B),B,B)
& m2_relset_1(k2_zfmodel1(A,B),B,B) ) ) ).
%------------------------------------------------------------------------------