SET007 Axioms: SET007+308.ax
%------------------------------------------------------------------------------
% File : SET007+308 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Category Ens
% 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 : ens_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 130 ( 5 unt; 0 def)
% Number of atoms : 663 ( 120 equ)
% Maximal formula atoms : 12 ( 5 avg)
% Number of connectives : 643 ( 110 ~; 1 |; 178 &)
% ( 20 <=>; 334 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 7 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 24 ( 22 usr; 1 prp; 0-3 aty)
% Number of functors : 67 ( 67 usr; 1 con; 0-6 aty)
% Number of variables : 341 ( 325 !; 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_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ v1_xboole_0(k1_ens_1(A))
& v1_fraenkel(k1_ens_1(A)) ) ) ).
fof(fc2_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ~ v1_xboole_0(k2_ens_1(A)) ) ).
fof(fc3_ens_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k2_ens_1(A)) )
=> ( v1_relat_1(k2_mcart_1(B))
& v1_funct_1(k2_mcart_1(B)) ) ) ).
fof(fc4_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_cat_1(k12_ens_1(A))
& v2_cat_1(k12_ens_1(A)) ) ) ).
fof(fc5_ens_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ~ v1_xboole_0(k17_ens_1(A)) ) ).
fof(t1_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( r2_hidden(B,k1_ens_1(A))
<=> ? [C] :
( m1_subset_1(C,A)
& ? [D] :
( m1_subset_1(D,A)
& ( D = k1_xboole_0
=> C = k1_xboole_0 )
& v1_funct_1(B)
& v1_funct_2(B,C,D)
& m2_relset_1(B,C,D) ) ) ) ) ).
fof(t2_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> r1_tarski(k1_funct_2(B,C),k1_ens_1(A)) ) ) ) ).
fof(t3_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> r1_tarski(k1_ens_1(B),k1_ens_1(A)) ) ) ).
fof(t4_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k2_ens_1(A))
=> ? [C] :
( m1_subset_1(C,k1_ens_1(A))
& ? [D] :
( m1_subset_1(D,A)
& ? [E] :
( m1_subset_1(E,A)
& B = k1_domain_1(k2_zfmisc_1(A,A),k1_ens_1(A),k1_domain_1(A,A,D,E),C)
& ( E = k1_xboole_0
=> D = k1_xboole_0 )
& v1_funct_1(C)
& v1_funct_2(C,D,E)
& m2_relset_1(C,D,E) ) ) ) ) ) ).
fof(t5_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,B,C)
& m2_relset_1(D,B,C) )
=> ( ( C = k1_xboole_0
=> B = k1_xboole_0 )
=> r2_hidden(k1_domain_1(k2_zfmisc_1(A,A),k1_zfmisc_1(k2_zfmisc_1(B,C)),k1_domain_1(A,A,B,C),D),k2_ens_1(A)) ) ) ) ) ) ).
fof(t6_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> r1_tarski(k2_ens_1(A),k2_zfmisc_1(k2_zfmisc_1(A,A),k1_ens_1(A))) ) ).
fof(t7_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> r1_tarski(k2_ens_1(B),k2_ens_1(A)) ) ) ).
fof(d3_ens_1,axiom,
$true ).
fof(d4_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k2_ens_1(A))
=> k3_ens_1(A,B) = k1_mcart_1(k1_mcart_1(B)) ) ) ).
fof(d5_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k2_ens_1(A))
=> k4_ens_1(A,B) = k2_mcart_1(k1_mcart_1(B)) ) ) ).
fof(t8_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k2_ens_1(A))
=> B = k4_tarski(k1_domain_1(A,A,k3_ens_1(A,B),k4_ens_1(A,B)),k2_mcart_1(B)) ) ) ).
fof(t9_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k2_ens_1(A))
=> ( ~ ( k4_ens_1(A,B) = k1_xboole_0
& k3_ens_1(A,B) != k1_xboole_0 )
& v1_funct_1(k2_mcart_1(B))
& v1_funct_2(k2_mcart_1(B),k3_ens_1(A,B),k4_ens_1(A,B))
& m2_relset_1(k2_mcart_1(B),k3_ens_1(A,B),k4_ens_1(A,B)) ) ) ) ).
fof(t10_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C,D] :
( r2_hidden(k4_tarski(k4_tarski(C,D),B),k2_ens_1(A))
=> ( ( D = k1_xboole_0
=> C = k1_xboole_0 )
& v1_funct_1(B)
& v1_funct_2(B,C,D)
& m2_relset_1(B,C,D) ) ) ) ) ).
fof(d6_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> k5_ens_1(A,B) = k1_domain_1(k2_zfmisc_1(A,A),k1_zfmisc_1(k2_zfmisc_1(B,B)),k1_domain_1(A,A,B,B),k6_partfun1(B)) ) ) ).
fof(t11_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ( k2_mcart_1(k5_ens_1(A,B)) = k6_partfun1(B)
& k3_ens_1(A,k5_ens_1(A,B)) = B
& k4_ens_1(A,k5_ens_1(A,B)) = B ) ) ) ).
fof(d7_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k2_ens_1(A))
=> ! [C] :
( m1_subset_1(C,k2_ens_1(A))
=> ( k4_ens_1(A,B) = k3_ens_1(A,C)
=> k6_ens_1(A,B,C) = k4_tarski(k1_domain_1(A,A,k3_ens_1(A,B),k4_ens_1(A,C)),k5_relat_1(k2_mcart_1(B),k2_mcart_1(C))) ) ) ) ) ).
fof(t12_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k2_ens_1(A))
=> ! [C] :
( m1_subset_1(C,k2_ens_1(A))
=> ( k3_ens_1(A,B) = k4_ens_1(A,C)
=> ( k2_mcart_1(k6_ens_1(A,C,B)) = k5_relat_1(k2_mcart_1(C),k2_mcart_1(B))
& k3_ens_1(A,k6_ens_1(A,C,B)) = k3_ens_1(A,C)
& k4_ens_1(A,k6_ens_1(A,C,B)) = k4_ens_1(A,B) ) ) ) ) ) ).
fof(t13_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k2_ens_1(A))
=> ! [C] :
( m1_subset_1(C,k2_ens_1(A))
=> ! [D] :
( m1_subset_1(D,k2_ens_1(A))
=> ( ( k3_ens_1(A,B) = k4_ens_1(A,C)
& k3_ens_1(A,D) = k4_ens_1(A,B) )
=> k6_ens_1(A,k6_ens_1(A,C,B),D) = k6_ens_1(A,C,k6_ens_1(A,B,D)) ) ) ) ) ) ).
fof(t14_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k2_ens_1(A))
=> ( k6_ens_1(A,k5_ens_1(A,k3_ens_1(A,B)),B) = B
& k6_ens_1(A,B,k5_ens_1(A,k4_ens_1(A,B))) = B ) ) ) ).
fof(t15_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,B,C)
& m2_relset_1(D,B,C) )
=> ( ( C = k1_xboole_0
=> B = k1_xboole_0 )
=> r2_hidden(k1_domain_1(k2_zfmisc_1(A,A),k1_zfmisc_1(k2_zfmisc_1(B,C)),k1_domain_1(A,A,B,C),D),k7_ens_1(A,B,C)) ) ) ) ) ) ).
fof(t16_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,k2_ens_1(A))
=> ( r2_hidden(D,k7_ens_1(A,B,C))
=> D = k4_tarski(k1_domain_1(A,A,B,C),k2_mcart_1(D)) ) ) ) ) ) ).
fof(t17_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> r1_tarski(k7_ens_1(A,B,C),k2_ens_1(A)) ) ) ) ).
fof(t19_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,k2_ens_1(A))
=> ( r2_hidden(D,k7_ens_1(A,B,C))
<=> ( k3_ens_1(A,D) = B
& k4_ens_1(A,D) = C ) ) ) ) ) ) ).
fof(t20_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,k2_ens_1(A))
=> ( r2_hidden(D,k7_ens_1(A,B,C))
=> r2_hidden(k2_mcart_1(D),k1_funct_2(B,C)) ) ) ) ) ) ).
fof(d9_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k2_ens_1(A))
=> ( v1_ens_1(B,A)
<=> k2_relat_1(k2_mcart_1(B)) = k4_ens_1(A,B) ) ) ) ).
fof(d10_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_ens_1(A),A)
& m2_relset_1(B,k2_ens_1(A),A) )
=> ( B = k8_ens_1(A)
<=> ! [C] :
( m1_subset_1(C,k2_ens_1(A))
=> k8_funct_2(k2_ens_1(A),A,B,C) = k3_ens_1(A,C) ) ) ) ) ).
fof(d11_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_ens_1(A),A)
& m2_relset_1(B,k2_ens_1(A),A) )
=> ( B = k9_ens_1(A)
<=> ! [C] :
( m1_subset_1(C,k2_ens_1(A))
=> k8_funct_2(k2_ens_1(A),A,B,C) = k4_ens_1(A,C) ) ) ) ) ).
fof(d12_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k2_zfmisc_1(k2_ens_1(A),k2_ens_1(A)),k2_ens_1(A)) )
=> ( B = k10_ens_1(A)
<=> ( ! [C] :
( m1_subset_1(C,k2_ens_1(A))
=> ! [D] :
( m1_subset_1(D,k2_ens_1(A))
=> ( r2_hidden(k1_domain_1(k2_ens_1(A),k2_ens_1(A),C,D),k1_relat_1(B))
<=> k3_ens_1(A,C) = k4_ens_1(A,D) ) ) )
& ! [C] :
( m1_subset_1(C,k2_ens_1(A))
=> ! [D] :
( m1_subset_1(D,k2_ens_1(A))
=> ( k3_ens_1(A,C) = k4_ens_1(A,D)
=> k1_funct_1(B,k1_domain_1(k2_ens_1(A),k2_ens_1(A),C,D)) = k6_ens_1(A,D,C) ) ) ) ) ) ) ) ).
fof(d13_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k2_ens_1(A))
& m2_relset_1(B,A,k2_ens_1(A)) )
=> ( B = k11_ens_1(A)
<=> ! [C] :
( m1_subset_1(C,A)
=> k8_funct_2(A,k2_ens_1(A),B,C) = k5_ens_1(A,C) ) ) ) ) ).
fof(d14_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k12_ens_1(A) = g1_cat_1(A,k2_ens_1(A),k8_ens_1(A),k9_ens_1(A),k10_ens_1(A),k11_ens_1(A)) ) ).
fof(t21_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v2_cat_1(g1_cat_1(A,k2_ens_1(A),k8_ens_1(A),k9_ens_1(A),k10_ens_1(A),k11_ens_1(A)))
& l1_cat_1(g1_cat_1(A,k2_ens_1(A),k8_ens_1(A),k9_ens_1(A),k10_ens_1(A),k11_ens_1(A))) ) ) ).
fof(t22_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> m1_subset_1(B,u1_cat_1(k12_ens_1(A))) ) ) ).
fof(d15_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> k13_ens_1(A,B) = B ) ) ).
fof(t23_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,u1_cat_1(k12_ens_1(A)))
=> m1_subset_1(B,A) ) ) ).
fof(d16_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,u1_cat_1(k12_ens_1(A)))
=> k14_ens_1(A,B) = B ) ) ).
fof(t24_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k2_ens_1(A))
=> m1_subset_1(B,u2_cat_1(k12_ens_1(A))) ) ) ).
fof(d17_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k2_ens_1(A))
=> k15_ens_1(A,B) = B ) ) ).
fof(t25_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,u2_cat_1(k12_ens_1(A)))
=> m1_subset_1(B,k2_ens_1(A)) ) ) ).
fof(d18_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,u2_cat_1(k12_ens_1(A)))
=> k16_ens_1(A,B) = B ) ) ).
fof(t26_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,u2_cat_1(k12_ens_1(A)))
=> ( k2_cat_1(k12_ens_1(A),B) = k3_ens_1(A,k16_ens_1(A,B))
& k3_cat_1(k12_ens_1(A),B) = k4_ens_1(A,k16_ens_1(A,B)) ) ) ) ).
fof(t27_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,u1_cat_1(k12_ens_1(A)))
=> ! [C] :
( m1_subset_1(C,u1_cat_1(k12_ens_1(A)))
=> k6_cat_1(k12_ens_1(A),B,C) = k7_ens_1(A,k14_ens_1(A,B),k14_ens_1(A,C)) ) ) ) ).
fof(t28_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,u2_cat_1(k12_ens_1(A)))
=> ! [C] :
( m1_subset_1(C,u2_cat_1(k12_ens_1(A)))
=> ( k2_cat_1(k12_ens_1(A),B) = k3_cat_1(k12_ens_1(A),C)
=> k4_cat_1(k12_ens_1(A),C,B) = k6_ens_1(A,k16_ens_1(A,C),k16_ens_1(A,B)) ) ) ) ) ).
fof(t29_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,u1_cat_1(k12_ens_1(A)))
=> k10_cat_1(k12_ens_1(A),B) = k5_ens_1(A,k14_ens_1(A,B)) ) ) ).
fof(t30_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,u1_cat_1(k12_ens_1(A)))
=> ( B = k1_xboole_0
=> v7_cat_1(B,k12_ens_1(A)) ) ) ) ).
fof(t31_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,u1_cat_1(k12_ens_1(A)))
=> ( ( r2_hidden(k1_xboole_0,A)
& v7_cat_1(B,k12_ens_1(A)) )
=> B = k1_xboole_0 ) ) ) ).
fof(t32_ens_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(k12_ens_1(A)))
=> ( v7_cat_1(B,k12_ens_1(A))
=> B = k1_xboole_0 ) ) ) ).
fof(t33_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,u1_cat_1(k12_ens_1(A)))
=> ( ? [C] : B = k1_tarski(C)
=> v6_cat_1(B,k12_ens_1(A)) ) ) ) ).
fof(t34_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,u1_cat_1(k12_ens_1(A)))
=> ~ ( A != k1_tarski(k1_xboole_0)
& v6_cat_1(B,k12_ens_1(A))
& ! [C] : B != k1_tarski(C) ) ) ) ).
fof(t35_ens_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(k12_ens_1(A)))
=> ~ ( v6_cat_1(B,k12_ens_1(A))
& ! [C] : B != k1_tarski(C) ) ) ) ).
fof(t36_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,u2_cat_1(k12_ens_1(A)))
=> ( v3_cat_1(B,k12_ens_1(A))
<=> v2_funct_1(k2_mcart_1(k16_ens_1(A,B))) ) ) ) ).
fof(t37_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,u2_cat_1(k12_ens_1(A)))
=> ( v4_cat_1(B,k12_ens_1(A))
=> ( ! [C] :
( m1_subset_1(C,A)
=> ! [D,E] :
~ ( r2_hidden(D,C)
& r2_hidden(E,C)
& D != E ) )
| v1_ens_1(k16_ens_1(A,B),A) ) ) ) ) ).
fof(t38_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,u2_cat_1(k12_ens_1(A)))
=> ( v1_ens_1(k16_ens_1(A,B),A)
=> v4_cat_1(B,k12_ens_1(A)) ) ) ) ).
fof(t39_ens_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m1_subset_1(B,u2_cat_1(k12_ens_1(A)))
=> ( v4_cat_1(B,k12_ens_1(A))
=> v1_ens_1(k16_ens_1(A,B),A) ) ) ) ).
fof(t40_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> r1_cat_2(k12_ens_1(B),k12_ens_1(A)) ) ) ).
fof(t41_ens_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u1_cat_1(A))
=> r2_hidden(k6_cat_1(A,B,C),k17_ens_1(A)) ) ) ) ).
fof(t42_ens_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u2_cat_1(A))
=> ( ( k6_cat_1(A,B,k3_cat_1(A,C)) = k1_xboole_0
=> k6_cat_1(A,B,k2_cat_1(A,C)) = k1_xboole_0 )
& ( k6_cat_1(A,k2_cat_1(A,C),B) = k1_xboole_0
=> k6_cat_1(A,k3_cat_1(A,C),B) = k1_xboole_0 ) ) ) ) ) ).
fof(d20_ens_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u2_cat_1(A))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k6_cat_1(A,B,k2_cat_1(A,C)),k6_cat_1(A,B,k3_cat_1(A,C)))
& m2_relset_1(D,k6_cat_1(A,B,k2_cat_1(A,C)),k6_cat_1(A,B,k3_cat_1(A,C))) )
=> ( D = k18_ens_1(A,B,C)
<=> ! [E] :
( m1_subset_1(E,u2_cat_1(A))
=> ( r2_hidden(E,k6_cat_1(A,B,k2_cat_1(A,C)))
=> k1_funct_1(D,E) = k4_cat_1(A,E,C) ) ) ) ) ) ) ) ).
fof(d21_ens_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u2_cat_1(A))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k6_cat_1(A,k3_cat_1(A,C),B),k6_cat_1(A,k2_cat_1(A,C),B))
& m2_relset_1(D,k6_cat_1(A,k3_cat_1(A,C),B),k6_cat_1(A,k2_cat_1(A,C),B)) )
=> ( D = k19_ens_1(A,B,C)
<=> ! [E] :
( m1_subset_1(E,u2_cat_1(A))
=> ( r2_hidden(E,k6_cat_1(A,k3_cat_1(A,C),B))
=> k1_funct_1(D,E) = k4_cat_1(A,C,E) ) ) ) ) ) ) ) ).
fof(t43_ens_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u1_cat_1(A))
=> k18_ens_1(A,B,k10_cat_1(A,C)) = k6_partfun1(k6_cat_1(A,B,C)) ) ) ) ).
fof(t44_ens_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u1_cat_1(A))
=> k19_ens_1(A,C,k10_cat_1(A,B)) = k6_partfun1(k6_cat_1(A,B,C)) ) ) ) ).
fof(t45_ens_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u2_cat_1(A))
=> ! [D] :
( m1_subset_1(D,u2_cat_1(A))
=> ( k2_cat_1(A,C) = k3_cat_1(A,D)
=> k18_ens_1(A,B,k4_cat_1(A,D,C)) = k1_partfun1(k6_cat_1(A,B,k2_cat_1(A,D)),k6_cat_1(A,B,k3_cat_1(A,D)),k6_cat_1(A,B,k2_cat_1(A,C)),k6_cat_1(A,B,k3_cat_1(A,C)),k18_ens_1(A,B,D),k18_ens_1(A,B,C)) ) ) ) ) ) ).
fof(t46_ens_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u2_cat_1(A))
=> ! [D] :
( m1_subset_1(D,u2_cat_1(A))
=> ( k2_cat_1(A,C) = k3_cat_1(A,D)
=> k19_ens_1(A,B,k4_cat_1(A,D,C)) = k1_partfun1(k6_cat_1(A,k3_cat_1(A,C),B),k6_cat_1(A,k2_cat_1(A,C),B),k6_cat_1(A,k3_cat_1(A,D),B),k6_cat_1(A,k2_cat_1(A,D),B),k19_ens_1(A,B,C),k19_ens_1(A,B,D)) ) ) ) ) ) ).
fof(t47_ens_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u2_cat_1(A))
=> m1_subset_1(k1_domain_1(k2_zfmisc_1(k1_zfmisc_1(u2_cat_1(A)),k1_zfmisc_1(u2_cat_1(A))),k1_zfmisc_1(k2_zfmisc_1(k6_cat_1(A,B,k2_cat_1(A,C)),k6_cat_1(A,B,k3_cat_1(A,C)))),k1_domain_1(k1_zfmisc_1(u2_cat_1(A)),k1_zfmisc_1(u2_cat_1(A)),k6_cat_1(A,B,k2_cat_1(A,C)),k6_cat_1(A,B,k3_cat_1(A,C))),k18_ens_1(A,B,C)),k2_ens_1(k17_ens_1(A))) ) ) ) ).
fof(t48_ens_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u2_cat_1(A))
=> m1_subset_1(k1_domain_1(k2_zfmisc_1(k1_zfmisc_1(u2_cat_1(A)),k1_zfmisc_1(u2_cat_1(A))),k1_zfmisc_1(k2_zfmisc_1(k6_cat_1(A,k3_cat_1(A,C),B),k6_cat_1(A,k2_cat_1(A,C),B))),k1_domain_1(k1_zfmisc_1(u2_cat_1(A)),k1_zfmisc_1(u2_cat_1(A)),k6_cat_1(A,k3_cat_1(A,C),B),k6_cat_1(A,k2_cat_1(A,C),B)),k19_ens_1(A,B,C)),k2_ens_1(k17_ens_1(A))) ) ) ) ).
fof(d22_ens_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u2_cat_1(A),k2_ens_1(k17_ens_1(A)))
& m2_relset_1(C,u2_cat_1(A),k2_ens_1(k17_ens_1(A))) )
=> ( C = k20_ens_1(A,B)
<=> ! [D] :
( m1_subset_1(D,u2_cat_1(A))
=> k8_funct_2(u2_cat_1(A),k2_ens_1(k17_ens_1(A)),C,D) = k1_domain_1(k2_zfmisc_1(k1_zfmisc_1(u2_cat_1(A)),k1_zfmisc_1(u2_cat_1(A))),k1_zfmisc_1(k2_zfmisc_1(k6_cat_1(A,B,k2_cat_1(A,D)),k6_cat_1(A,B,k3_cat_1(A,D)))),k1_domain_1(k1_zfmisc_1(u2_cat_1(A)),k1_zfmisc_1(u2_cat_1(A)),k6_cat_1(A,B,k2_cat_1(A,D)),k6_cat_1(A,B,k3_cat_1(A,D))),k18_ens_1(A,B,D)) ) ) ) ) ) ).
fof(d23_ens_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u2_cat_1(A),k2_ens_1(k17_ens_1(A)))
& m2_relset_1(C,u2_cat_1(A),k2_ens_1(k17_ens_1(A))) )
=> ( C = k21_ens_1(A,B)
<=> ! [D] :
( m1_subset_1(D,u2_cat_1(A))
=> k8_funct_2(u2_cat_1(A),k2_ens_1(k17_ens_1(A)),C,D) = k1_domain_1(k2_zfmisc_1(k1_zfmisc_1(u2_cat_1(A)),k1_zfmisc_1(u2_cat_1(A))),k1_zfmisc_1(k2_zfmisc_1(k6_cat_1(A,k3_cat_1(A,D),B),k6_cat_1(A,k2_cat_1(A,D),B))),k1_domain_1(k1_zfmisc_1(u2_cat_1(A)),k1_zfmisc_1(u2_cat_1(A)),k6_cat_1(A,k3_cat_1(A,D),B),k6_cat_1(A,k2_cat_1(A,D),B)),k19_ens_1(A,B,D)) ) ) ) ) ) ).
fof(t49_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_cat_1(B))
=> ( r1_tarski(k17_ens_1(B),A)
=> m2_cat_1(k20_ens_1(B,C),B,k12_ens_1(A)) ) ) ) ) ).
fof(t50_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_cat_1(B))
=> ( r1_tarski(k17_ens_1(B),A)
=> m1_oppcat_1(k21_ens_1(B,C),B,k12_ens_1(A)) ) ) ) ) ).
fof(t51_ens_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u2_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u2_cat_1(A))
=> ( k6_cat_1(A,k2_cat_1(A,B),k3_cat_1(A,C)) = k1_xboole_0
=> k6_cat_1(A,k3_cat_1(A,B),k2_cat_1(A,C)) = k1_xboole_0 ) ) ) ) ).
fof(d24_ens_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u2_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u2_cat_1(A))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k6_cat_1(A,k3_cat_1(A,B),k2_cat_1(A,C)),k6_cat_1(A,k2_cat_1(A,B),k3_cat_1(A,C)))
& m2_relset_1(D,k6_cat_1(A,k3_cat_1(A,B),k2_cat_1(A,C)),k6_cat_1(A,k2_cat_1(A,B),k3_cat_1(A,C))) )
=> ( D = k22_ens_1(A,B,C)
<=> ! [E] :
( m1_subset_1(E,u2_cat_1(A))
=> ( r2_hidden(E,k6_cat_1(A,k3_cat_1(A,B),k2_cat_1(A,C)))
=> k1_funct_1(D,E) = k4_cat_1(A,B,k4_cat_1(A,E,C)) ) ) ) ) ) ) ) ).
fof(t52_ens_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u2_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u2_cat_1(A))
=> m1_subset_1(k1_domain_1(k2_zfmisc_1(k1_zfmisc_1(u2_cat_1(A)),k1_zfmisc_1(u2_cat_1(A))),k1_zfmisc_1(k2_zfmisc_1(k6_cat_1(A,k3_cat_1(A,B),k2_cat_1(A,C)),k6_cat_1(A,k2_cat_1(A,B),k3_cat_1(A,C)))),k1_domain_1(k1_zfmisc_1(u2_cat_1(A)),k1_zfmisc_1(u2_cat_1(A)),k6_cat_1(A,k3_cat_1(A,B),k2_cat_1(A,C)),k6_cat_1(A,k2_cat_1(A,B),k3_cat_1(A,C))),k22_ens_1(A,B,C)),k2_ens_1(k17_ens_1(A))) ) ) ) ).
fof(t53_ens_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u2_cat_1(A))
=> ( k22_ens_1(A,k10_cat_1(A,B),C) = k18_ens_1(A,B,C)
& k22_ens_1(A,C,k10_cat_1(A,B)) = k19_ens_1(A,B,C) ) ) ) ) ).
fof(t54_ens_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u1_cat_1(A))
=> k22_ens_1(A,k10_cat_1(A,B),k10_cat_1(A,C)) = k6_partfun1(k6_cat_1(A,B,C)) ) ) ) ).
fof(t55_ens_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u2_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u2_cat_1(A))
=> k22_ens_1(A,B,C) = k1_partfun1(k6_cat_1(A,k3_cat_1(A,B),k2_cat_1(A,C)),k6_cat_1(A,k2_cat_1(A,B),k2_cat_1(A,C)),k6_cat_1(A,k2_cat_1(A,B),k2_cat_1(A,C)),k6_cat_1(A,k2_cat_1(A,B),k3_cat_1(A,C)),k19_ens_1(A,k2_cat_1(A,C),B),k18_ens_1(A,k2_cat_1(A,B),C)) ) ) ) ).
fof(t56_ens_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u2_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u2_cat_1(A))
=> ! [D] :
( m1_subset_1(D,u2_cat_1(A))
=> ! [E] :
( m1_subset_1(E,u2_cat_1(A))
=> ( ( k3_cat_1(A,B) = k2_cat_1(A,C)
& k2_cat_1(A,D) = k3_cat_1(A,E) )
=> k22_ens_1(A,k4_cat_1(A,B,C),k4_cat_1(A,E,D)) = k1_partfun1(k6_cat_1(A,k3_cat_1(A,C),k2_cat_1(A,E)),k6_cat_1(A,k2_cat_1(A,C),k3_cat_1(A,E)),k6_cat_1(A,k3_cat_1(A,B),k2_cat_1(A,D)),k6_cat_1(A,k2_cat_1(A,B),k3_cat_1(A,D)),k22_ens_1(A,C,E),k22_ens_1(A,B,D)) ) ) ) ) ) ) ).
fof(d25_ens_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u2_cat_1(k11_cat_2(A,A)),k2_ens_1(k17_ens_1(A)))
& m2_relset_1(B,u2_cat_1(k11_cat_2(A,A)),k2_ens_1(k17_ens_1(A))) )
=> ( B = k23_ens_1(A)
<=> ! [C] :
( m1_subset_1(C,u2_cat_1(A))
=> ! [D] :
( m1_subset_1(D,u2_cat_1(A))
=> k8_funct_2(u2_cat_1(k11_cat_2(A,A)),k2_ens_1(k17_ens_1(A)),B,k13_cat_2(A,A,C,D)) = k1_domain_1(k2_zfmisc_1(k1_zfmisc_1(u2_cat_1(A)),k1_zfmisc_1(u2_cat_1(A))),k1_zfmisc_1(k2_zfmisc_1(k6_cat_1(A,k3_cat_1(A,C),k2_cat_1(A,D)),k6_cat_1(A,k2_cat_1(A,C),k3_cat_1(A,D)))),k1_domain_1(k1_zfmisc_1(u2_cat_1(A)),k1_zfmisc_1(u2_cat_1(A)),k6_cat_1(A,k3_cat_1(A,C),k2_cat_1(A,D)),k6_cat_1(A,k2_cat_1(A,C),k3_cat_1(A,D))),k22_ens_1(A,C,D)) ) ) ) ) ) ).
fof(t57_ens_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ( k20_ens_1(A,B) = k1_funct_1(k3_funct_5(k23_ens_1(A)),k10_cat_1(A,B))
& k21_ens_1(A,B) = k1_funct_1(k5_funct_5(k23_ens_1(A)),k10_cat_1(A,B)) ) ) ) ).
fof(t58_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ( r1_tarski(k17_ens_1(B),A)
=> m2_cat_1(k23_ens_1(B),k11_cat_2(k2_oppcat_1(B),B),k12_ens_1(A)) ) ) ) ).
fof(d26_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_cat_1(B))
=> ( r1_tarski(k17_ens_1(B),A)
=> k24_ens_1(A,B,C) = k20_ens_1(B,C) ) ) ) ) ).
fof(d27_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_cat_1(B))
=> ( r1_tarski(k17_ens_1(B),A)
=> k25_ens_1(A,B,C) = k21_ens_1(B,C) ) ) ) ) ).
fof(d28_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ( r1_tarski(k17_ens_1(B),A)
=> k26_ens_1(A,B) = k23_ens_1(B) ) ) ) ).
fof(t59_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_cat_1(B))
=> ! [D] :
( m1_subset_1(D,u2_cat_1(B))
=> ( r1_tarski(k17_ens_1(B),A)
=> k8_funct_2(u2_cat_1(B),u2_cat_1(k12_ens_1(A)),k24_ens_1(A,B,C),D) = k1_domain_1(k2_zfmisc_1(k1_zfmisc_1(u2_cat_1(B)),k1_zfmisc_1(u2_cat_1(B))),k1_zfmisc_1(k2_zfmisc_1(k6_cat_1(B,C,k2_cat_1(B,D)),k6_cat_1(B,C,k3_cat_1(B,D)))),k1_domain_1(k1_zfmisc_1(u2_cat_1(B)),k1_zfmisc_1(u2_cat_1(B)),k6_cat_1(B,C,k2_cat_1(B,D)),k6_cat_1(B,C,k3_cat_1(B,D))),k18_ens_1(B,C,D)) ) ) ) ) ) ).
fof(t60_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_cat_1(B))
=> ! [D] :
( m1_subset_1(D,u1_cat_1(B))
=> ( r1_tarski(k17_ens_1(B),A)
=> k8_funct_2(u1_cat_1(B),u1_cat_1(k12_ens_1(A)),k12_cat_1(B,k12_ens_1(A),k24_ens_1(A,B,C)),D) = k6_cat_1(B,C,D) ) ) ) ) ) ).
fof(t61_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_cat_1(B))
=> ! [D] :
( m1_subset_1(D,u2_cat_1(B))
=> ( r1_tarski(k17_ens_1(B),A)
=> k8_funct_2(u2_cat_1(B),u2_cat_1(k12_ens_1(A)),k25_ens_1(A,B,C),D) = k1_domain_1(k2_zfmisc_1(k1_zfmisc_1(u2_cat_1(B)),k1_zfmisc_1(u2_cat_1(B))),k1_zfmisc_1(k2_zfmisc_1(k6_cat_1(B,k3_cat_1(B,D),C),k6_cat_1(B,k2_cat_1(B,D),C))),k1_domain_1(k1_zfmisc_1(u2_cat_1(B)),k1_zfmisc_1(u2_cat_1(B)),k6_cat_1(B,k3_cat_1(B,D),C),k6_cat_1(B,k2_cat_1(B,D),C)),k19_ens_1(B,C,D)) ) ) ) ) ) ).
fof(t62_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_cat_1(B))
=> ! [D] :
( m1_subset_1(D,u1_cat_1(B))
=> ( r1_tarski(k17_ens_1(B),A)
=> k8_funct_2(u1_cat_1(B),u1_cat_1(k12_ens_1(A)),k12_cat_1(B,k12_ens_1(A),k25_ens_1(A,B,C)),D) = k6_cat_1(B,D,C) ) ) ) ) ) ).
fof(t63_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u2_cat_1(B))
=> ! [D] :
( m1_subset_1(D,u2_cat_1(B))
=> ( r1_tarski(k17_ens_1(B),A)
=> k8_funct_2(u2_cat_1(k11_cat_2(k2_oppcat_1(B),B)),u2_cat_1(k12_ens_1(A)),k26_ens_1(A,B),k13_cat_2(k2_oppcat_1(B),B,k5_oppcat_1(B,C),D)) = k1_domain_1(k2_zfmisc_1(k1_zfmisc_1(u2_cat_1(B)),k1_zfmisc_1(u2_cat_1(B))),k1_zfmisc_1(k2_zfmisc_1(k6_cat_1(B,k3_cat_1(B,C),k2_cat_1(B,D)),k6_cat_1(B,k2_cat_1(B,C),k3_cat_1(B,D)))),k1_domain_1(k1_zfmisc_1(u2_cat_1(B)),k1_zfmisc_1(u2_cat_1(B)),k6_cat_1(B,k3_cat_1(B,C),k2_cat_1(B,D)),k6_cat_1(B,k2_cat_1(B,C),k3_cat_1(B,D))),k22_ens_1(B,C,D)) ) ) ) ) ) ).
fof(t64_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_cat_1(B))
=> ! [D] :
( m1_subset_1(D,u1_cat_1(B))
=> ( r1_tarski(k17_ens_1(B),A)
=> k8_funct_2(u1_cat_1(k11_cat_2(k2_oppcat_1(B),B)),u1_cat_1(k12_ens_1(A)),k12_cat_1(k11_cat_2(k2_oppcat_1(B),B),k12_ens_1(A),k26_ens_1(A,B)),k12_cat_2(k2_oppcat_1(B),B,k3_oppcat_1(B,C),D)) = k6_cat_1(B,C,D) ) ) ) ) ) ).
fof(t65_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_cat_1(B))
=> ( r1_tarski(k17_ens_1(B),A)
=> k14_cat_2(k2_oppcat_1(B),B,k12_ens_1(A),k26_ens_1(A,B),k3_oppcat_1(B,C)) = k24_ens_1(A,B,C) ) ) ) ) ).
fof(t66_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_cat_1(B))
=> ( r1_tarski(k17_ens_1(B),A)
=> k15_cat_2(k2_oppcat_1(B),B,k12_ens_1(A),k26_ens_1(A,B),C) = k25_ens_1(A,B,C) ) ) ) ) ).
fof(dt_k1_ens_1,axiom,
$true ).
fof(dt_k2_ens_1,axiom,
$true ).
fof(dt_k3_ens_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k2_ens_1(A)) )
=> m1_subset_1(k3_ens_1(A,B),A) ) ).
fof(dt_k4_ens_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k2_ens_1(A)) )
=> m1_subset_1(k4_ens_1(A,B),A) ) ).
fof(dt_k5_ens_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A) )
=> m1_subset_1(k5_ens_1(A,B),k2_ens_1(A)) ) ).
fof(dt_k6_ens_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k2_ens_1(A))
& m1_subset_1(C,k2_ens_1(A)) )
=> m1_subset_1(k6_ens_1(A,B,C),k2_ens_1(A)) ) ).
fof(dt_k7_ens_1,axiom,
$true ).
fof(dt_k8_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_funct_1(k8_ens_1(A))
& v1_funct_2(k8_ens_1(A),k2_ens_1(A),A)
& m2_relset_1(k8_ens_1(A),k2_ens_1(A),A) ) ) ).
fof(dt_k9_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_funct_1(k9_ens_1(A))
& v1_funct_2(k9_ens_1(A),k2_ens_1(A),A)
& m2_relset_1(k9_ens_1(A),k2_ens_1(A),A) ) ) ).
fof(dt_k10_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_funct_1(k10_ens_1(A))
& m2_relset_1(k10_ens_1(A),k2_zfmisc_1(k2_ens_1(A),k2_ens_1(A)),k2_ens_1(A)) ) ) ).
fof(dt_k11_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_funct_1(k11_ens_1(A))
& v1_funct_2(k11_ens_1(A),A,k2_ens_1(A))
& m2_relset_1(k11_ens_1(A),A,k2_ens_1(A)) ) ) ).
fof(dt_k12_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> l1_cat_1(k12_ens_1(A)) ) ).
fof(dt_k13_ens_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A) )
=> m1_subset_1(k13_ens_1(A,B),u1_cat_1(k12_ens_1(A))) ) ).
fof(dt_k14_ens_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,u1_cat_1(k12_ens_1(A))) )
=> m1_subset_1(k14_ens_1(A,B),A) ) ).
fof(dt_k15_ens_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k2_ens_1(A)) )
=> m1_subset_1(k15_ens_1(A,B),u2_cat_1(k12_ens_1(A))) ) ).
fof(dt_k16_ens_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,u2_cat_1(k12_ens_1(A))) )
=> m1_subset_1(k16_ens_1(A,B),k2_ens_1(A)) ) ).
fof(dt_k17_ens_1,axiom,
$true ).
fof(dt_k18_ens_1,axiom,
! [A,B,C] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A))
& m1_subset_1(C,u2_cat_1(A)) )
=> ( v1_funct_1(k18_ens_1(A,B,C))
& v1_funct_2(k18_ens_1(A,B,C),k6_cat_1(A,B,k2_cat_1(A,C)),k6_cat_1(A,B,k3_cat_1(A,C)))
& m2_relset_1(k18_ens_1(A,B,C),k6_cat_1(A,B,k2_cat_1(A,C)),k6_cat_1(A,B,k3_cat_1(A,C))) ) ) ).
fof(dt_k19_ens_1,axiom,
! [A,B,C] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A))
& m1_subset_1(C,u2_cat_1(A)) )
=> ( v1_funct_1(k19_ens_1(A,B,C))
& v1_funct_2(k19_ens_1(A,B,C),k6_cat_1(A,k3_cat_1(A,C),B),k6_cat_1(A,k2_cat_1(A,C),B))
& m2_relset_1(k19_ens_1(A,B,C),k6_cat_1(A,k3_cat_1(A,C),B),k6_cat_1(A,k2_cat_1(A,C),B)) ) ) ).
fof(dt_k20_ens_1,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A)) )
=> ( v1_funct_1(k20_ens_1(A,B))
& v1_funct_2(k20_ens_1(A,B),u2_cat_1(A),k2_ens_1(k17_ens_1(A)))
& m2_relset_1(k20_ens_1(A,B),u2_cat_1(A),k2_ens_1(k17_ens_1(A))) ) ) ).
fof(dt_k21_ens_1,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A)) )
=> ( v1_funct_1(k21_ens_1(A,B))
& v1_funct_2(k21_ens_1(A,B),u2_cat_1(A),k2_ens_1(k17_ens_1(A)))
& m2_relset_1(k21_ens_1(A,B),u2_cat_1(A),k2_ens_1(k17_ens_1(A))) ) ) ).
fof(dt_k22_ens_1,axiom,
! [A,B,C] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u2_cat_1(A))
& m1_subset_1(C,u2_cat_1(A)) )
=> ( v1_funct_1(k22_ens_1(A,B,C))
& v1_funct_2(k22_ens_1(A,B,C),k6_cat_1(A,k3_cat_1(A,B),k2_cat_1(A,C)),k6_cat_1(A,k2_cat_1(A,B),k3_cat_1(A,C)))
& m2_relset_1(k22_ens_1(A,B,C),k6_cat_1(A,k3_cat_1(A,B),k2_cat_1(A,C)),k6_cat_1(A,k2_cat_1(A,B),k3_cat_1(A,C))) ) ) ).
fof(dt_k23_ens_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ( v1_funct_1(k23_ens_1(A))
& v1_funct_2(k23_ens_1(A),u2_cat_1(k11_cat_2(A,A)),k2_ens_1(k17_ens_1(A)))
& m2_relset_1(k23_ens_1(A),u2_cat_1(k11_cat_2(A,A)),k2_ens_1(k17_ens_1(A))) ) ) ).
fof(dt_k24_ens_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m1_subset_1(C,u1_cat_1(B)) )
=> m2_cat_1(k24_ens_1(A,B,C),B,k12_ens_1(A)) ) ).
fof(dt_k25_ens_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m1_subset_1(C,u1_cat_1(B)) )
=> m1_oppcat_1(k25_ens_1(A,B,C),B,k12_ens_1(A)) ) ).
fof(dt_k26_ens_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> m2_cat_1(k26_ens_1(A,B),k11_cat_2(k2_oppcat_1(B),B),k12_ens_1(A)) ) ).
fof(d1_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k1_ens_1(A) = k3_tarski(a_1_0_ens_1(A)) ) ).
fof(d2_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k2_ens_1(A) = a_1_1_ens_1(A) ) ).
fof(d8_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> k7_ens_1(A,B,C) = a_3_0_ens_1(A,B,C) ) ) ) ).
fof(t18_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k2_ens_1(A) = k3_tarski(a_1_2_ens_1(A)) ) ).
fof(d19_ens_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> k17_ens_1(A) = a_1_3_ens_1(A) ) ).
fof(fraenkel_a_1_0_ens_1,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ( r2_hidden(A,a_1_0_ens_1(B))
<=> ? [C,D] :
( m1_subset_1(C,B)
& m1_subset_1(D,B)
& A = k1_funct_2(C,D) ) ) ) ).
fof(fraenkel_a_1_1_ens_1,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ( r2_hidden(A,a_1_1_ens_1(B))
<=> ? [C,D,E] :
( m1_subset_1(C,B)
& m1_subset_1(D,B)
& m1_subset_1(E,k1_ens_1(B))
& A = k1_domain_1(k2_zfmisc_1(B,B),k1_ens_1(B),k1_domain_1(B,B,C,D),E)
& ( D = k1_xboole_0
=> C = k1_xboole_0 )
& v1_funct_1(E)
& v1_funct_2(E,C,D)
& m2_relset_1(E,C,D) ) ) ) ).
fof(fraenkel_a_3_0_ens_1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(C,B)
& m1_subset_1(D,B) )
=> ( r2_hidden(A,a_3_0_ens_1(B,C,D))
<=> ? [E] :
( m1_subset_1(E,k1_ens_1(B))
& A = k1_domain_1(k2_zfmisc_1(B,B),k1_ens_1(B),k1_domain_1(B,B,C,D),E)
& r2_hidden(k1_domain_1(k2_zfmisc_1(B,B),k1_ens_1(B),k1_domain_1(B,B,C,D),E),k2_ens_1(B)) ) ) ) ).
fof(fraenkel_a_1_2_ens_1,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ( r2_hidden(A,a_1_2_ens_1(B))
<=> ? [C,D] :
( m1_subset_1(C,B)
& m1_subset_1(D,B)
& A = k7_ens_1(B,C,D) ) ) ) ).
fof(fraenkel_a_1_3_ens_1,axiom,
! [A,B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ( r2_hidden(A,a_1_3_ens_1(B))
<=> ? [C,D] :
( m1_subset_1(C,u1_cat_1(B))
& m1_subset_1(D,u1_cat_1(B))
& A = k6_cat_1(B,C,D) ) ) ) ).
%------------------------------------------------------------------------------