SET007 Axioms: SET007+57.ax
%------------------------------------------------------------------------------
% File : SET007+57 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Models and Satisfiability
% 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 : zf_model [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 60 ( 9 unt; 0 def)
% Number of atoms : 598 ( 96 equ)
% Maximal formula atoms : 45 ( 9 avg)
% Number of connectives : 634 ( 96 ~; 1 |; 284 &)
% ( 37 <=>; 216 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 10 avg)
% Maximal term depth : 11 ( 1 avg)
% Number of predicates : 23 ( 21 usr; 1 prp; 0-3 aty)
% Number of functors : 80 ( 80 usr; 18 con; 0-4 aty)
% Number of variables : 234 ( 218 !; 16 ?)
% 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(fc1_zf_model,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ~ v1_xboole_0(k3_zf_model(A)) ) ).
fof(d1_zf_model,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( B = k1_zf_model(A)
<=> ? [C] :
( ! [D] :
( m2_subset_1(D,k5_numbers,k1_zf_lang)
=> ! [E] :
( m2_subset_1(E,k5_numbers,k1_zf_lang)
=> ( r2_hidden(k4_tarski(k4_zf_lang(D,E),k2_tarski(D,E)),C)
& r2_hidden(k4_tarski(k5_zf_lang(D,E),k2_tarski(D,E)),C) ) ) )
& r2_hidden(k4_tarski(A,B),C)
& ! [D] :
( ( v1_zf_lang(D)
& m2_finseq_1(D,k5_numbers) )
=> ! [E] :
( r2_hidden(k4_tarski(D,E),C)
=> ( ( v2_zf_lang(D)
=> E = k2_tarski(k18_zf_lang(D),k19_zf_lang(D)) )
& ( v3_zf_lang(D)
=> E = k2_tarski(k18_zf_lang(D),k19_zf_lang(D)) )
& ~ ( v4_zf_lang(D)
& ! [F] :
~ ( E = F
& r2_hidden(k4_tarski(k20_zf_lang(D),F),C) ) )
& ~ ( v5_zf_lang(D)
& ! [F,G] :
~ ( E = k3_tarski(k2_tarski(F,G))
& r2_hidden(k4_tarski(k21_zf_lang(D),F),C)
& r2_hidden(k4_tarski(k22_zf_lang(D),G),C) ) )
& ~ ( v6_zf_lang(D)
& ! [F] :
~ ( E = k4_xboole_0(k3_tarski(k1_tarski(F)),k1_tarski(k23_zf_lang(D)))
& r2_hidden(k4_tarski(k24_zf_lang(D),F),C) ) ) ) ) ) ) ) ) ).
fof(t1_zf_model,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( ( v2_zf_lang(A)
=> k2_zf_model(A) = k2_tarski(k18_zf_lang(A),k19_zf_lang(A)) )
& ( v3_zf_lang(A)
=> k2_zf_model(A) = k2_tarski(k18_zf_lang(A),k19_zf_lang(A)) )
& ( v4_zf_lang(A)
=> k2_zf_model(A) = k2_zf_model(k20_zf_lang(A)) )
& ( v5_zf_lang(A)
=> k2_zf_model(A) = k4_subset_1(k1_zf_lang,k2_zf_model(k21_zf_lang(A)),k2_zf_model(k22_zf_lang(A))) )
& ( v6_zf_lang(A)
=> k2_zf_model(A) = k4_xboole_0(k2_zf_model(k24_zf_lang(A)),k1_tarski(k23_zf_lang(A))) ) ) ) ).
fof(d2_zf_model,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( B = k3_zf_model(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> ( v1_funct_1(C)
& v1_funct_2(C,k1_zf_lang,A)
& m2_relset_1(C,k1_zf_lang,A) ) ) ) ) ).
fof(t2_zf_model,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ! [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k1_zf_lang,A)
& m2_relset_1(D,k1_zf_lang,A) )
=> ( k8_funct_2(k1_zf_lang,A,D,B) = k8_funct_2(k1_zf_lang,A,D,C)
<=> r2_hidden(D,k5_zf_model(k4_zf_lang(B,C),A)) ) ) ) ) ) ).
fof(t3_zf_model,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ! [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k1_zf_lang,A)
& m2_relset_1(D,k1_zf_lang,A) )
=> ( r2_hidden(k8_funct_2(k1_zf_lang,A,D,B),k8_funct_2(k1_zf_lang,A,D,C))
<=> r2_hidden(D,k5_zf_model(k5_zf_lang(B,C),A)) ) ) ) ) ) ).
fof(t4_zf_model,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) )
=> ( ~ r2_hidden(C,k5_zf_model(B,A))
<=> r2_hidden(C,k5_zf_model(k6_zf_lang(B),A)) ) ) ) ) ).
fof(t5_zf_model,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k1_zf_lang,A)
& m2_relset_1(D,k1_zf_lang,A) )
=> ( ( r2_hidden(D,k5_zf_model(B,A))
& r2_hidden(D,k5_zf_model(C,A)) )
<=> r2_hidden(D,k5_zf_model(k7_zf_lang(B,C),A)) ) ) ) ) ) ).
fof(t6_zf_model,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k1_zf_lang,A)
& m2_relset_1(D,k1_zf_lang,A) )
=> ( ( r2_hidden(D,k5_zf_model(C,A))
& ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k1_zf_lang,A)
& m2_relset_1(E,k1_zf_lang,A) )
=> ( ! [F] :
( m2_subset_1(F,k5_numbers,k1_zf_lang)
=> ( k8_funct_2(k1_zf_lang,A,E,F) != k8_funct_2(k1_zf_lang,A,D,F)
=> B = F ) )
=> r2_hidden(E,k5_zf_model(C,A)) ) ) )
<=> r2_hidden(D,k5_zf_model(k8_zf_lang(B,C),A)) ) ) ) ) ) ).
fof(t7_zf_model,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ~ v1_xboole_0(B)
=> ( v2_zf_lang(A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_zf_lang,B)
& m2_relset_1(C,k1_zf_lang,B) )
=> ( k8_funct_2(k1_zf_lang,B,C,k18_zf_lang(A)) = k8_funct_2(k1_zf_lang,B,C,k19_zf_lang(A))
<=> r2_hidden(C,k5_zf_model(A,B)) ) ) ) ) ) ).
fof(t8_zf_model,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ~ v1_xboole_0(B)
=> ( v3_zf_lang(A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_zf_lang,B)
& m2_relset_1(C,k1_zf_lang,B) )
=> ( r2_hidden(k8_funct_2(k1_zf_lang,B,C,k18_zf_lang(A)),k8_funct_2(k1_zf_lang,B,C,k19_zf_lang(A)))
<=> r2_hidden(C,k5_zf_model(A,B)) ) ) ) ) ) ).
fof(t9_zf_model,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ~ v1_xboole_0(B)
=> ( v4_zf_lang(A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_zf_lang,B)
& m2_relset_1(C,k1_zf_lang,B) )
=> ( ~ r2_hidden(C,k5_zf_model(k20_zf_lang(A),B))
<=> r2_hidden(C,k5_zf_model(A,B)) ) ) ) ) ) ).
fof(t10_zf_model,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ~ v1_xboole_0(B)
=> ( v5_zf_lang(A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_zf_lang,B)
& m2_relset_1(C,k1_zf_lang,B) )
=> ( ( r2_hidden(C,k5_zf_model(k21_zf_lang(A),B))
& r2_hidden(C,k5_zf_model(k22_zf_lang(A),B)) )
<=> r2_hidden(C,k5_zf_model(A,B)) ) ) ) ) ) ).
fof(t11_zf_model,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ~ v1_xboole_0(B)
=> ( v6_zf_lang(A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_zf_lang,B)
& m2_relset_1(C,k1_zf_lang,B) )
=> ( ( r2_hidden(C,k5_zf_model(k24_zf_lang(A),B))
& ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k1_zf_lang,B)
& m2_relset_1(D,k1_zf_lang,B) )
=> ( ! [E] :
( m2_subset_1(E,k5_numbers,k1_zf_lang)
=> ( k8_funct_2(k1_zf_lang,B,D,E) != k8_funct_2(k1_zf_lang,B,C,E)
=> k23_zf_lang(A) = E ) )
=> r2_hidden(D,k5_zf_model(k24_zf_lang(A),B)) ) ) )
<=> r2_hidden(C,k5_zf_model(A,B)) ) ) ) ) ) ).
fof(d4_zf_model,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_zf_lang,A)
& m2_relset_1(B,k1_zf_lang,A) )
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ( r1_zf_model(A,B,C)
<=> r2_hidden(B,k5_zf_model(C,A)) ) ) ) ) ).
fof(t12_zf_model,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_zf_lang,A)
& m2_relset_1(B,k1_zf_lang,A) )
=> ! [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
=> ! [D] :
( m2_subset_1(D,k5_numbers,k1_zf_lang)
=> ( r1_zf_model(A,B,k4_zf_lang(C,D))
<=> k8_funct_2(k1_zf_lang,A,B,C) = k8_funct_2(k1_zf_lang,A,B,D) ) ) ) ) ) ).
fof(t13_zf_model,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_zf_lang,A)
& m2_relset_1(B,k1_zf_lang,A) )
=> ! [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
=> ! [D] :
( m2_subset_1(D,k5_numbers,k1_zf_lang)
=> ( r1_zf_model(A,B,k5_zf_lang(C,D))
<=> r2_hidden(k8_funct_2(k1_zf_lang,A,B,C),k8_funct_2(k1_zf_lang,A,B,D)) ) ) ) ) ) ).
fof(t14_zf_model,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_zf_lang,A)
& m2_relset_1(B,k1_zf_lang,A) )
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ( r1_zf_model(A,B,C)
<=> ~ r1_zf_model(A,B,k6_zf_lang(C)) ) ) ) ) ).
fof(t15_zf_model,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_zf_lang,A)
& m2_relset_1(B,k1_zf_lang,A) )
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ! [D] :
( ( v1_zf_lang(D)
& m2_finseq_1(D,k5_numbers) )
=> ( r1_zf_model(A,B,k7_zf_lang(C,D))
<=> ( r1_zf_model(A,B,C)
& r1_zf_model(A,B,D) ) ) ) ) ) ) ).
fof(t16_zf_model,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_zf_lang,A)
& m2_relset_1(B,k1_zf_lang,A) )
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ! [D] :
( m2_subset_1(D,k5_numbers,k1_zf_lang)
=> ( r1_zf_model(A,B,k8_zf_lang(D,C))
<=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k1_zf_lang,A)
& m2_relset_1(E,k1_zf_lang,A) )
=> ( ! [F] :
( m2_subset_1(F,k5_numbers,k1_zf_lang)
=> ( k8_funct_2(k1_zf_lang,A,E,F) != k8_funct_2(k1_zf_lang,A,B,F)
=> D = F ) )
=> r1_zf_model(A,E,C) ) ) ) ) ) ) ) ).
fof(t17_zf_model,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_zf_lang,A)
& m2_relset_1(B,k1_zf_lang,A) )
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ! [D] :
( ( v1_zf_lang(D)
& m2_finseq_1(D,k5_numbers) )
=> ( r1_zf_model(A,B,k10_zf_lang(C,D))
<=> ( r1_zf_model(A,B,C)
| r1_zf_model(A,B,D) ) ) ) ) ) ) ).
fof(t18_zf_model,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_zf_lang,A)
& m2_relset_1(B,k1_zf_lang,A) )
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ! [D] :
( ( v1_zf_lang(D)
& m2_finseq_1(D,k5_numbers) )
=> ( r1_zf_model(A,B,k11_zf_lang(C,D))
<=> ( r1_zf_model(A,B,C)
=> r1_zf_model(A,B,D) ) ) ) ) ) ) ).
fof(t19_zf_model,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_zf_lang,A)
& m2_relset_1(B,k1_zf_lang,A) )
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ! [D] :
( ( v1_zf_lang(D)
& m2_finseq_1(D,k5_numbers) )
=> ( r1_zf_model(A,B,k12_zf_lang(C,D))
<=> ( r1_zf_model(A,B,C)
<=> r1_zf_model(A,B,D) ) ) ) ) ) ) ).
fof(t20_zf_model,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_zf_lang,A)
& m2_relset_1(B,k1_zf_lang,A) )
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ! [D] :
( m2_subset_1(D,k5_numbers,k1_zf_lang)
=> ( r1_zf_model(A,B,k13_zf_lang(D,C))
<=> ? [E] :
( v1_funct_1(E)
& v1_funct_2(E,k1_zf_lang,A)
& m2_relset_1(E,k1_zf_lang,A)
& ! [F] :
( m2_subset_1(F,k5_numbers,k1_zf_lang)
=> ( k8_funct_2(k1_zf_lang,A,E,F) != k8_funct_2(k1_zf_lang,A,B,F)
=> D = F ) )
& r1_zf_model(A,E,C) ) ) ) ) ) ) ).
fof(t21_zf_model,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_zf_lang,A)
& m2_relset_1(B,k1_zf_lang,A) )
=> ! [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
=> ! [D] :
( m1_subset_1(D,A)
=> ? [E] :
( v1_funct_1(E)
& v1_funct_2(E,k1_zf_lang,A)
& m2_relset_1(E,k1_zf_lang,A)
& k8_funct_2(k1_zf_lang,A,E,C) = D
& ! [F] :
( m2_subset_1(F,k5_numbers,k1_zf_lang)
=> ( F != C
=> k8_funct_2(k1_zf_lang,A,E,F) = k8_funct_2(k1_zf_lang,A,B,F) ) ) ) ) ) ) ) ).
fof(t22_zf_model,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ! [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
=> ! [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_zf_model(D,E,k14_zf_lang(B,C,A))
<=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k1_zf_lang,D)
& m2_relset_1(F,k1_zf_lang,D) )
=> ( ! [G] :
( m2_subset_1(G,k5_numbers,k1_zf_lang)
=> ~ ( k8_funct_2(k1_zf_lang,D,F,G) != k8_funct_2(k1_zf_lang,D,E,G)
& B != G
& C != G ) )
=> r1_zf_model(D,F,A) ) ) ) ) ) ) ) ) ).
fof(t23_zf_model,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ! [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
=> ! [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_zf_model(D,E,k15_zf_lang(B,C,A))
<=> ? [F] :
( v1_funct_1(F)
& v1_funct_2(F,k1_zf_lang,D)
& m2_relset_1(F,k1_zf_lang,D)
& ! [G] :
( m2_subset_1(G,k5_numbers,k1_zf_lang)
=> ~ ( k8_funct_2(k1_zf_lang,D,F,G) != k8_funct_2(k1_zf_lang,D,E,G)
& B != G
& C != G ) )
& r1_zf_model(D,F,A) ) ) ) ) ) ) ) ).
fof(d5_zf_model,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ( r2_zf_model(A,B)
<=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_zf_lang,A)
& m2_relset_1(C,k1_zf_lang,A) )
=> r1_zf_model(A,C,B) ) ) ) ) ).
fof(t24_zf_model,axiom,
$true ).
fof(t25_zf_model,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ( r2_zf_model(C,k8_zf_lang(B,A))
<=> r2_zf_model(C,A) ) ) ) ) ).
fof(d6_zf_model,axiom,
k6_zf_model = k14_zf_lang(k2_zf_lang(np__0),k2_zf_lang(np__1),k11_zf_lang(k8_zf_lang(k2_zf_lang(np__2),k12_zf_lang(k5_zf_lang(k2_zf_lang(np__2),k2_zf_lang(np__0)),k5_zf_lang(k2_zf_lang(np__2),k2_zf_lang(np__1)))),k4_zf_lang(k2_zf_lang(np__0),k2_zf_lang(np__1)))) ).
fof(d7_zf_model,axiom,
k7_zf_model = k14_zf_lang(k2_zf_lang(np__0),k2_zf_lang(np__1),k13_zf_lang(k2_zf_lang(np__2),k8_zf_lang(k2_zf_lang(np__3),k12_zf_lang(k5_zf_lang(k2_zf_lang(np__3),k2_zf_lang(np__2)),k10_zf_lang(k4_zf_lang(k2_zf_lang(np__3),k2_zf_lang(np__0)),k4_zf_lang(k2_zf_lang(np__3),k2_zf_lang(np__1))))))) ).
fof(d8_zf_model,axiom,
k8_zf_model = k8_zf_lang(k2_zf_lang(np__0),k13_zf_lang(k2_zf_lang(np__1),k8_zf_lang(k2_zf_lang(np__2),k12_zf_lang(k5_zf_lang(k2_zf_lang(np__2),k2_zf_lang(np__1)),k13_zf_lang(k2_zf_lang(np__3),k7_zf_lang(k5_zf_lang(k2_zf_lang(np__2),k2_zf_lang(np__3)),k5_zf_lang(k2_zf_lang(np__3),k2_zf_lang(np__0)))))))) ).
fof(d9_zf_model,axiom,
k9_zf_model = k15_zf_lang(k2_zf_lang(np__0),k2_zf_lang(np__1),k7_zf_lang(k5_zf_lang(k2_zf_lang(np__1),k2_zf_lang(np__0)),k8_zf_lang(k2_zf_lang(np__2),k11_zf_lang(k5_zf_lang(k2_zf_lang(np__2),k2_zf_lang(np__0)),k13_zf_lang(k2_zf_lang(np__3),k7_zf_lang(k7_zf_lang(k5_zf_lang(k2_zf_lang(np__3),k2_zf_lang(np__0)),k6_zf_lang(k4_zf_lang(k2_zf_lang(np__3),k2_zf_lang(np__2)))),k8_zf_lang(k2_zf_lang(np__4),k11_zf_lang(k5_zf_lang(k2_zf_lang(np__4),k2_zf_lang(np__2)),k5_zf_lang(k2_zf_lang(np__4),k2_zf_lang(np__3)))))))))) ).
fof(d10_zf_model,axiom,
k10_zf_model = k8_zf_lang(k2_zf_lang(np__0),k13_zf_lang(k2_zf_lang(np__1),k8_zf_lang(k2_zf_lang(np__2),k12_zf_lang(k5_zf_lang(k2_zf_lang(np__2),k2_zf_lang(np__1)),k8_zf_lang(k2_zf_lang(np__3),k11_zf_lang(k5_zf_lang(k2_zf_lang(np__3),k2_zf_lang(np__2)),k5_zf_lang(k2_zf_lang(np__3),k2_zf_lang(np__0)))))))) ).
fof(d11_zf_model,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> k11_zf_model(A) = k11_zf_lang(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)))))),k8_zf_lang(k2_zf_lang(np__1),k13_zf_lang(k2_zf_lang(np__2),k8_zf_lang(k2_zf_lang(np__4),k12_zf_lang(k5_zf_lang(k2_zf_lang(np__4),k2_zf_lang(np__2)),k13_zf_lang(k2_zf_lang(np__3),k7_zf_lang(k5_zf_lang(k2_zf_lang(np__3),k2_zf_lang(np__1)),A))))))) ) ).
fof(d12_zf_model,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_zf_model(A)
<=> ( v1_ordinal1(A)
& r2_zf_model(A,k7_zf_model)
& r2_zf_model(A,k8_zf_model)
& r2_zf_model(A,k9_zf_model)
& r2_zf_model(A,k10_zf_model)
& ! [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)) ) ) ) ) ) ).
fof(s1_zf_model,axiom,
? [A,B] :
( ! [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
=> ! [D] :
( m2_subset_1(D,k5_numbers,k1_zf_lang)
=> ( r2_hidden(k4_tarski(k4_zf_lang(C,D),f1_s1_zf_model(C,D)),B)
& r2_hidden(k4_tarski(k5_zf_lang(C,D),f2_s1_zf_model(C,D)),B) ) ) )
& r2_hidden(k4_tarski(f6_s1_zf_model,A),B)
& ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ! [D] :
( r2_hidden(k4_tarski(C,D),B)
=> ( ( v2_zf_lang(C)
=> D = f1_s1_zf_model(k18_zf_lang(C),k19_zf_lang(C)) )
& ( v3_zf_lang(C)
=> D = f2_s1_zf_model(k18_zf_lang(C),k19_zf_lang(C)) )
& ~ ( v4_zf_lang(C)
& ! [E] :
~ ( D = f3_s1_zf_model(E)
& r2_hidden(k4_tarski(k20_zf_lang(C),E),B) ) )
& ~ ( v5_zf_lang(C)
& ! [E,F] :
~ ( D = f4_s1_zf_model(E,F)
& r2_hidden(k4_tarski(k21_zf_lang(C),E),B)
& r2_hidden(k4_tarski(k22_zf_lang(C),F),B) ) )
& ~ ( v6_zf_lang(C)
& ! [E] :
~ ( D = f5_s1_zf_model(k23_zf_lang(C),E)
& r2_hidden(k4_tarski(k24_zf_lang(C),E),B) ) ) ) ) ) ) ).
fof(s2_zf_model,axiom,
( ( ? [A] :
( ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ! [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
=> ( r2_hidden(k4_tarski(k4_zf_lang(B,C),f1_s2_zf_model(B,C)),A)
& r2_hidden(k4_tarski(k5_zf_lang(B,C),f2_s2_zf_model(B,C)),A) ) ) )
& r2_hidden(k4_tarski(f6_s2_zf_model,f7_s2_zf_model),A)
& ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ! [C] :
( r2_hidden(k4_tarski(B,C),A)
=> ( ( v2_zf_lang(B)
=> C = f1_s2_zf_model(k18_zf_lang(B),k19_zf_lang(B)) )
& ( v3_zf_lang(B)
=> C = f2_s2_zf_model(k18_zf_lang(B),k19_zf_lang(B)) )
& ~ ( v4_zf_lang(B)
& ! [D] :
~ ( C = f3_s2_zf_model(D)
& r2_hidden(k4_tarski(k20_zf_lang(B),D),A) ) )
& ~ ( v5_zf_lang(B)
& ! [D,E] :
~ ( C = f4_s2_zf_model(D,E)
& r2_hidden(k4_tarski(k21_zf_lang(B),D),A)
& r2_hidden(k4_tarski(k22_zf_lang(B),E),A) ) )
& ~ ( v6_zf_lang(B)
& ! [D] :
~ ( C = f5_s2_zf_model(k23_zf_lang(B),D)
& r2_hidden(k4_tarski(k24_zf_lang(B),D),A) ) ) ) ) ) )
& ? [A] :
( ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ! [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
=> ( r2_hidden(k4_tarski(k4_zf_lang(B,C),f1_s2_zf_model(B,C)),A)
& r2_hidden(k4_tarski(k5_zf_lang(B,C),f2_s2_zf_model(B,C)),A) ) ) )
& r2_hidden(k4_tarski(f6_s2_zf_model,f8_s2_zf_model),A)
& ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ! [C] :
( r2_hidden(k4_tarski(B,C),A)
=> ( ( v2_zf_lang(B)
=> C = f1_s2_zf_model(k18_zf_lang(B),k19_zf_lang(B)) )
& ( v3_zf_lang(B)
=> C = f2_s2_zf_model(k18_zf_lang(B),k19_zf_lang(B)) )
& ~ ( v4_zf_lang(B)
& ! [D] :
~ ( C = f3_s2_zf_model(D)
& r2_hidden(k4_tarski(k20_zf_lang(B),D),A) ) )
& ~ ( v5_zf_lang(B)
& ! [D,E] :
~ ( C = f4_s2_zf_model(D,E)
& r2_hidden(k4_tarski(k21_zf_lang(B),D),A)
& r2_hidden(k4_tarski(k22_zf_lang(B),E),A) ) )
& ~ ( v6_zf_lang(B)
& ! [D] :
~ ( C = f5_s2_zf_model(k23_zf_lang(B),D)
& r2_hidden(k4_tarski(k24_zf_lang(B),D),A) ) ) ) ) ) ) )
=> f7_s2_zf_model = f8_s2_zf_model ) ).
fof(s3_zf_model,axiom,
( ! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( B = f7_s3_zf_model(A)
<=> ? [C] :
( ! [D] :
( m2_subset_1(D,k5_numbers,k1_zf_lang)
=> ! [E] :
( m2_subset_1(E,k5_numbers,k1_zf_lang)
=> ( r2_hidden(k4_tarski(k4_zf_lang(D,E),f1_s3_zf_model(D,E)),C)
& r2_hidden(k4_tarski(k5_zf_lang(D,E),f2_s3_zf_model(D,E)),C) ) ) )
& r2_hidden(k4_tarski(A,B),C)
& ! [D] :
( ( v1_zf_lang(D)
& m2_finseq_1(D,k5_numbers) )
=> ! [E] :
( r2_hidden(k4_tarski(D,E),C)
=> ( ( v2_zf_lang(D)
=> E = f1_s3_zf_model(k18_zf_lang(D),k19_zf_lang(D)) )
& ( v3_zf_lang(D)
=> E = f2_s3_zf_model(k18_zf_lang(D),k19_zf_lang(D)) )
& ~ ( v4_zf_lang(D)
& ! [F] :
~ ( E = f3_s3_zf_model(F)
& r2_hidden(k4_tarski(k20_zf_lang(D),F),C) ) )
& ~ ( v5_zf_lang(D)
& ! [F,G] :
~ ( E = f4_s3_zf_model(F,G)
& r2_hidden(k4_tarski(k21_zf_lang(D),F),C)
& r2_hidden(k4_tarski(k22_zf_lang(D),G),C) ) )
& ~ ( v6_zf_lang(D)
& ! [F] :
~ ( E = f5_s3_zf_model(k23_zf_lang(D),F)
& r2_hidden(k4_tarski(k24_zf_lang(D),F),C) ) ) ) ) ) ) ) )
=> ( ( v2_zf_lang(f6_s3_zf_model)
=> f7_s3_zf_model(f6_s3_zf_model) = f1_s3_zf_model(k18_zf_lang(f6_s3_zf_model),k19_zf_lang(f6_s3_zf_model)) )
& ( v3_zf_lang(f6_s3_zf_model)
=> f7_s3_zf_model(f6_s3_zf_model) = f2_s3_zf_model(k18_zf_lang(f6_s3_zf_model),k19_zf_lang(f6_s3_zf_model)) )
& ( v4_zf_lang(f6_s3_zf_model)
=> f7_s3_zf_model(f6_s3_zf_model) = f3_s3_zf_model(f7_s3_zf_model(k20_zf_lang(f6_s3_zf_model))) )
& ( v5_zf_lang(f6_s3_zf_model)
=> ! [A,B] :
( ( A = f7_s3_zf_model(k21_zf_lang(f6_s3_zf_model))
& B = f7_s3_zf_model(k22_zf_lang(f6_s3_zf_model)) )
=> f7_s3_zf_model(f6_s3_zf_model) = f4_s3_zf_model(A,B) ) )
& ( v6_zf_lang(f6_s3_zf_model)
=> f7_s3_zf_model(f6_s3_zf_model) = f5_s3_zf_model(k23_zf_lang(f6_s3_zf_model),f7_s3_zf_model(k24_zf_lang(f6_s3_zf_model))) ) ) ) ).
fof(s4_zf_model,axiom,
( ( ! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( B = f6_s4_zf_model(A)
<=> ? [C] :
( ! [D] :
( m2_subset_1(D,k5_numbers,k1_zf_lang)
=> ! [E] :
( m2_subset_1(E,k5_numbers,k1_zf_lang)
=> ( r2_hidden(k4_tarski(k4_zf_lang(D,E),f1_s4_zf_model(D,E)),C)
& r2_hidden(k4_tarski(k5_zf_lang(D,E),f2_s4_zf_model(D,E)),C) ) ) )
& r2_hidden(k4_tarski(A,B),C)
& ! [D] :
( ( v1_zf_lang(D)
& m2_finseq_1(D,k5_numbers) )
=> ! [E] :
( r2_hidden(k4_tarski(D,E),C)
=> ( ( v2_zf_lang(D)
=> E = f1_s4_zf_model(k18_zf_lang(D),k19_zf_lang(D)) )
& ( v3_zf_lang(D)
=> E = f2_s4_zf_model(k18_zf_lang(D),k19_zf_lang(D)) )
& ~ ( v4_zf_lang(D)
& ! [F] :
~ ( E = f3_s4_zf_model(F)
& r2_hidden(k4_tarski(k20_zf_lang(D),F),C) ) )
& ~ ( v5_zf_lang(D)
& ! [F,G] :
~ ( E = f4_s4_zf_model(F,G)
& r2_hidden(k4_tarski(k21_zf_lang(D),F),C)
& r2_hidden(k4_tarski(k22_zf_lang(D),G),C) ) )
& ~ ( v6_zf_lang(D)
& ! [F] :
~ ( E = f5_s4_zf_model(k23_zf_lang(D),F)
& r2_hidden(k4_tarski(k24_zf_lang(D),F),C) ) ) ) ) ) ) ) )
& ! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ( p1_s4_zf_model(f1_s4_zf_model(A,B))
& p1_s4_zf_model(f2_s4_zf_model(A,B)) ) ) )
& ! [A] :
( p1_s4_zf_model(A)
=> p1_s4_zf_model(f3_s4_zf_model(A)) )
& ! [A,B] :
( ( p1_s4_zf_model(A)
& p1_s4_zf_model(B) )
=> p1_s4_zf_model(f4_s4_zf_model(A,B)) )
& ! [A,B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ( p1_s4_zf_model(A)
=> p1_s4_zf_model(f5_s4_zf_model(B,A)) ) ) )
=> p1_s4_zf_model(f6_s4_zf_model(f7_s4_zf_model)) ) ).
fof(dt_k1_zf_model,axiom,
$true ).
fof(dt_k2_zf_model,axiom,
! [A] :
( ( v1_zf_lang(A)
& m1_finseq_1(A,k5_numbers) )
=> m1_subset_1(k2_zf_model(A),k1_zfmisc_1(k1_zf_lang)) ) ).
fof(redefinition_k2_zf_model,axiom,
! [A] :
( ( v1_zf_lang(A)
& m1_finseq_1(A,k5_numbers) )
=> k2_zf_model(A) = k1_zf_model(A) ) ).
fof(dt_k3_zf_model,axiom,
$true ).
fof(dt_k4_zf_model,axiom,
$true ).
fof(dt_k5_zf_model,axiom,
! [A,B] :
( ( v1_zf_lang(A)
& m1_finseq_1(A,k5_numbers)
& ~ v1_xboole_0(B) )
=> m1_subset_1(k5_zf_model(A,B),k1_zfmisc_1(k3_zf_model(B))) ) ).
fof(redefinition_k5_zf_model,axiom,
! [A,B] :
( ( v1_zf_lang(A)
& m1_finseq_1(A,k5_numbers)
& ~ v1_xboole_0(B) )
=> k5_zf_model(A,B) = k4_zf_model(A,B) ) ).
fof(dt_k6_zf_model,axiom,
( v1_zf_lang(k6_zf_model)
& m2_finseq_1(k6_zf_model,k5_numbers) ) ).
fof(dt_k7_zf_model,axiom,
( v1_zf_lang(k7_zf_model)
& m2_finseq_1(k7_zf_model,k5_numbers) ) ).
fof(dt_k8_zf_model,axiom,
( v1_zf_lang(k8_zf_model)
& m2_finseq_1(k8_zf_model,k5_numbers) ) ).
fof(dt_k9_zf_model,axiom,
( v1_zf_lang(k9_zf_model)
& m2_finseq_1(k9_zf_model,k5_numbers) ) ).
fof(dt_k10_zf_model,axiom,
( v1_zf_lang(k10_zf_model)
& m2_finseq_1(k10_zf_model,k5_numbers) ) ).
fof(dt_k11_zf_model,axiom,
! [A] :
( ( v1_zf_lang(A)
& m1_finseq_1(A,k5_numbers) )
=> ( v1_zf_lang(k11_zf_model(A))
& m2_finseq_1(k11_zf_model(A),k5_numbers) ) ) ).
fof(d3_zf_model,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( C = k4_zf_model(A,B)
<=> ? [D] :
( ! [E] :
( m2_subset_1(E,k5_numbers,k1_zf_lang)
=> ! [F] :
( m2_subset_1(F,k5_numbers,k1_zf_lang)
=> ( r2_hidden(k4_tarski(k4_zf_lang(E,F),a_3_0_zf_model(B,E,F)),D)
& r2_hidden(k4_tarski(k5_zf_lang(E,F),a_3_1_zf_model(B,E,F)),D) ) ) )
& r2_hidden(k4_tarski(A,C),D)
& ! [E] :
( ( v1_zf_lang(E)
& m2_finseq_1(E,k5_numbers) )
=> ! [F] :
( r2_hidden(k4_tarski(E,F),D)
=> ( ( v2_zf_lang(E)
=> F = a_2_0_zf_model(B,E) )
& ( v3_zf_lang(E)
=> F = a_2_1_zf_model(B,E) )
& ~ ( v4_zf_lang(E)
& ! [G] :
~ ( F = k4_xboole_0(k3_zf_model(B),k3_tarski(k1_tarski(G)))
& r2_hidden(k4_tarski(k20_zf_lang(E),G),D) ) )
& ~ ( v5_zf_lang(E)
& ! [G,H] :
~ ( F = k3_xboole_0(k3_tarski(k1_tarski(G)),k3_tarski(k1_tarski(H)))
& r2_hidden(k4_tarski(k21_zf_lang(E),G),D)
& r2_hidden(k4_tarski(k22_zf_lang(E),H),D) ) )
& ~ ( v6_zf_lang(E)
& ! [G] :
~ ( F = a_3_2_zf_model(B,E,G)
& r2_hidden(k4_tarski(k24_zf_lang(E),G),D) ) ) ) ) ) ) ) ) ) ).
fof(fraenkel_a_3_0_zf_model,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(C,k5_numbers,k1_zf_lang)
& m2_subset_1(D,k5_numbers,k1_zf_lang) )
=> ( r2_hidden(A,a_3_0_zf_model(B,C,D))
<=> ? [E] :
( m1_subset_1(E,k3_zf_model(B))
& A = E
& ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k1_zf_lang,B)
& m2_relset_1(F,k1_zf_lang,B) )
=> ( F = E
=> k8_funct_2(k1_zf_lang,B,F,C) = k8_funct_2(k1_zf_lang,B,F,D) ) ) ) ) ) ).
fof(fraenkel_a_3_1_zf_model,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(C,k5_numbers,k1_zf_lang)
& m2_subset_1(D,k5_numbers,k1_zf_lang) )
=> ( r2_hidden(A,a_3_1_zf_model(B,C,D))
<=> ? [E] :
( m1_subset_1(E,k3_zf_model(B))
& A = E
& ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k1_zf_lang,B)
& m2_relset_1(F,k1_zf_lang,B) )
=> ( F = E
=> r2_hidden(k8_funct_2(k1_zf_lang,B,F,C),k8_funct_2(k1_zf_lang,B,F,D)) ) ) ) ) ) ).
fof(fraenkel_a_2_0_zf_model,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ( r2_hidden(A,a_2_0_zf_model(B,C))
<=> ? [D] :
( m1_subset_1(D,k3_zf_model(B))
& A = D
& ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k1_zf_lang,B)
& m2_relset_1(E,k1_zf_lang,B) )
=> ( E = D
=> k8_funct_2(k1_zf_lang,B,E,k18_zf_lang(C)) = k8_funct_2(k1_zf_lang,B,E,k19_zf_lang(C)) ) ) ) ) ) ).
fof(fraenkel_a_2_1_zf_model,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ( r2_hidden(A,a_2_1_zf_model(B,C))
<=> ? [D] :
( m1_subset_1(D,k3_zf_model(B))
& A = D
& ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k1_zf_lang,B)
& m2_relset_1(E,k1_zf_lang,B) )
=> ( E = D
=> r2_hidden(k8_funct_2(k1_zf_lang,B,E,k18_zf_lang(C)),k8_funct_2(k1_zf_lang,B,E,k19_zf_lang(C))) ) ) ) ) ) ).
fof(fraenkel_a_3_2_zf_model,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ( r2_hidden(A,a_3_2_zf_model(B,C,D))
<=> ? [E] :
( m1_subset_1(E,k3_zf_model(B))
& A = E
& ! [F,G] :
( ( v1_funct_1(G)
& v1_funct_2(G,k1_zf_lang,B)
& m2_relset_1(G,k1_zf_lang,B) )
=> ( ( F = D
& G = E )
=> ( r2_hidden(G,F)
& ! [H] :
( ( v1_funct_1(H)
& v1_funct_2(H,k1_zf_lang,B)
& m2_relset_1(H,k1_zf_lang,B) )
=> ( ! [I] :
( m2_subset_1(I,k5_numbers,k1_zf_lang)
=> ( k8_funct_2(k1_zf_lang,B,H,I) != k8_funct_2(k1_zf_lang,B,G,I)
=> k23_zf_lang(C) = I ) )
=> r2_hidden(H,F) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------