SET007 Axioms: SET007+291.ax
%------------------------------------------------------------------------------
% File : SET007+291 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Algebra of Normal Forms Is a Heyting Algebra
% 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 : heyting1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 65 ( 11 unt; 0 def)
% Number of atoms : 246 ( 42 equ)
% Maximal formula atoms : 14 ( 3 avg)
% Number of connectives : 201 ( 20 ~; 0 |; 68 &)
% ( 11 <=>; 102 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 6 avg)
% Maximal term depth : 6 ( 2 avg)
% Number of predicates : 27 ( 25 usr; 1 prp; 0-4 aty)
% Number of functors : 50 ( 50 usr; 1 con; 0-6 aty)
% Number of variables : 159 ( 154 !; 5 ?)
% 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(rc1_heyting1,axiom,
! [A] :
? [B] :
( m1_subset_1(B,k8_normform(A))
& ~ v1_xboole_0(B) ) ).
fof(fc1_heyting1,axiom,
! [A] :
( ~ v3_struct_0(k12_normform(A))
& v3_lattices(k12_normform(A))
& v4_lattices(k12_normform(A))
& v5_lattices(k12_normform(A))
& v6_lattices(k12_normform(A))
& v7_lattices(k12_normform(A))
& v8_lattices(k12_normform(A))
& v9_lattices(k12_normform(A))
& v10_lattices(k12_normform(A))
& v11_lattices(k12_normform(A))
& v12_lattices(k12_normform(A))
& v13_lattices(k12_normform(A))
& v3_filter_0(k12_normform(A)) ) ).
fof(t1_heyting1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) )
=> ( ! [E] :
( m1_subset_1(E,A)
=> r2_hidden(k8_funct_2(A,B,D,E),C) )
=> ( v1_funct_1(D)
& v1_funct_2(D,A,C)
& m2_relset_1(D,A,C) ) ) ) ) ) ) ).
fof(d1_heyting1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k5_finsub_1(A))
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(A))
=> ( r1_heyting1(A,B,C)
<=> ! [D] :
( m1_subset_1(D,A)
=> ( r2_hidden(D,B)
=> r2_hidden(D,C) ) ) ) ) ) ) ).
fof(d2_heyting1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k1_heyting1(A) = A ) ).
fof(t2_heyting1,axiom,
$true ).
fof(t3_heyting1,axiom,
! [A,B] :
( m1_subset_1(B,k5_finsub_1(k7_normform(A)))
=> ( B = k1_xboole_0
=> k9_normform(A,B) = k1_xboole_0 ) ) ).
fof(d3_heyting1,axiom,
! [A,B] :
( m1_subset_1(B,u1_struct_0(k12_normform(A)))
=> k3_heyting1(A,B) = B ) ).
fof(t4_heyting1,axiom,
$true ).
fof(t5_heyting1,axiom,
$true ).
fof(t6_heyting1,axiom,
$true ).
fof(t7_heyting1,axiom,
! [A,B] :
( m2_subset_1(B,k5_finsub_1(k7_normform(A)),k8_normform(A))
=> k9_normform(A,k10_normform(A,B,B)) = B ) ).
fof(t8_heyting1,axiom,
! [A,B] :
( m2_subset_1(B,k5_finsub_1(k7_normform(A)),k8_normform(A))
=> ! [C] :
( r1_tarski(C,B)
=> r2_hidden(C,k8_normform(A)) ) ) ).
fof(t9_heyting1,axiom,
$true ).
fof(t10_heyting1,axiom,
! [A,B] :
( m1_subset_1(B,u1_struct_0(k12_normform(A)))
=> ! [C] :
( r1_tarski(C,B)
=> m1_subset_1(C,u1_struct_0(k12_normform(A))) ) ) ).
fof(d4_heyting1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k7_normform(A),u1_struct_0(k12_normform(A)))
& m2_relset_1(B,k7_normform(A),u1_struct_0(k12_normform(A))) )
=> ( B = k4_heyting1(A)
<=> ! [C] :
( m2_subset_1(C,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> k8_funct_2(k7_normform(A),u1_struct_0(k12_normform(A)),B,C) = k2_heyting1(A,C) ) ) ) ).
fof(t11_heyting1,axiom,
! [A,B] :
( m2_subset_1(B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ! [C] :
( m2_subset_1(C,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ( r2_hidden(B,k8_funct_2(k7_normform(A),u1_struct_0(k12_normform(A)),k4_heyting1(A),C))
=> B = C ) ) ) ).
fof(t12_heyting1,axiom,
! [A,B] :
( m2_subset_1(B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> r2_hidden(B,k8_funct_2(k7_normform(A),u1_struct_0(k12_normform(A)),k4_heyting1(A),B)) ) ).
fof(t13_heyting1,axiom,
! [A,B] :
( m2_subset_1(B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> k8_funct_2(k7_normform(A),u1_struct_0(k12_normform(A)),k4_heyting1(A),B) = k8_funct_2(k7_normform(A),k5_finsub_1(k7_normform(A)),k11_setwiseo(k7_normform(A)),B) ) ).
fof(t14_heyting1,axiom,
! [A,B] :
( m2_subset_1(B,k5_finsub_1(k7_normform(A)),k8_normform(A))
=> k2_lattice2(k7_normform(A),k12_normform(A),B,k4_heyting1(A)) = k10_setwiseo(k7_normform(A),k7_normform(A),B,k11_setwiseo(k7_normform(A))) ) ).
fof(t15_heyting1,axiom,
! [A,B] :
( m1_subset_1(B,u1_struct_0(k12_normform(A)))
=> B = k2_lattice2(k7_normform(A),k12_normform(A),k3_heyting1(A,B),k4_heyting1(A)) ) ).
fof(d5_heyting1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A))),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)))
& m2_relset_1(B,k2_zfmisc_1(k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A))),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A))) )
=> ( B = k5_heyting1(A)
<=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)))
=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)))
=> k2_binop_1(k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),B,C,D) = k3_normform(k5_finsub_1(A),k5_finsub_1(A),C,D) ) ) ) ) ).
fof(t17_heyting1,axiom,
! [A] : m2_subset_1(k4_tarski(k1_xboole_0,k1_xboole_0),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A)) ).
fof(t18_heyting1,axiom,
! [A,B] :
( m2_subset_1(B,k5_finsub_1(k7_normform(A)),k8_normform(A))
=> ( B = k1_xboole_0
=> k6_heyting1(A,B) = k1_tarski(k4_tarski(k1_xboole_0,k1_xboole_0)) ) ) ).
fof(t19_heyting1,axiom,
! [A,B] :
( m2_subset_1(B,k5_finsub_1(k7_normform(A)),k8_normform(A))
=> ! [C] :
( m2_subset_1(C,k5_finsub_1(k7_normform(A)),k8_normform(A))
=> ( ( B = k1_xboole_0
& C = k1_xboole_0 )
=> k7_heyting1(A,B,C) = k1_tarski(k4_tarski(k1_xboole_0,k1_xboole_0)) ) ) ) ).
fof(t20_heyting1,axiom,
! [A] :
( m2_subset_1(A,k2_zfmisc_1(k5_finsub_1(k1_xboole_0),k5_finsub_1(k1_xboole_0)),k7_normform(k1_xboole_0))
=> A = k4_tarski(k1_xboole_0,k1_xboole_0) ) ).
fof(t21_heyting1,axiom,
k7_normform(k1_xboole_0) = k1_tarski(k4_tarski(k1_xboole_0,k1_xboole_0)) ).
fof(t22_heyting1,axiom,
! [A] : m2_subset_1(k1_tarski(k4_tarski(k1_xboole_0,k1_xboole_0)),k5_finsub_1(k7_normform(A)),k8_normform(A)) ).
fof(t23_heyting1,axiom,
! [A,B] :
( m1_subset_1(B,k5_finsub_1(k7_normform(A)))
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(k7_normform(A)))
=> ! [D] :
( m2_subset_1(D,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ~ ( r2_hidden(D,k7_heyting1(A,B,C))
& ! [E] :
( m2_fraenkel(E,k7_normform(A),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k1_fraenkel(k7_normform(A),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A))))
=> ~ ( r1_tarski(k8_setwiseo(k7_normform(A),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),E,B),C)
& D = k6_normform(k7_normform(A),A,B,k6_funcop_1(k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A),k5_heyting1(A),E,k6_funct_3(k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A)))) ) ) ) ) ) ) ).
fof(t24_heyting1,axiom,
! [A,B] :
( m2_subset_1(B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ! [C] :
( m2_subset_1(C,k5_finsub_1(k7_normform(A)),k8_normform(A))
=> ~ ( k10_normform(A,C,k2_heyting1(A,B)) = k1_xboole_0
& ! [D] :
( m2_subset_1(D,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ~ ( r2_hidden(D,k6_heyting1(A,C))
& r1_normform(k5_finsub_1(A),k5_finsub_1(A),D,B) ) ) ) ) ) ).
fof(t25_heyting1,axiom,
! [A,B] :
( m2_subset_1(B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k12_normform(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k12_normform(A)))
=> ~ ( ! [E] :
( m2_subset_1(E,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ( r2_hidden(E,C)
=> r2_hidden(k1_normform(k5_finsub_1(A),k5_finsub_1(A),E,B),k7_normform(A)) ) )
& ! [E] :
( m2_subset_1(E,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ~ ( r2_hidden(E,C)
& ! [F] :
( m2_subset_1(F,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ~ ( r2_hidden(F,D)
& r1_normform(k5_finsub_1(A),k5_finsub_1(A),F,k1_normform(k5_finsub_1(A),k5_finsub_1(A),E,B)) ) ) ) )
& ! [E] :
( m2_subset_1(E,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ~ ( r2_hidden(E,k7_heyting1(A,k3_heyting1(A,C),k3_heyting1(A,D)))
& r1_normform(k5_finsub_1(A),k5_finsub_1(A),E,B) ) ) ) ) ) ) ).
fof(t26_heyting1,axiom,
! [A,B] :
( m2_subset_1(B,k5_finsub_1(k7_normform(A)),k8_normform(A))
=> k10_normform(A,B,k6_heyting1(A,B)) = k1_xboole_0 ) ).
fof(d8_heyting1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k12_normform(A)),u1_struct_0(k12_normform(A)))
& m2_relset_1(B,u1_struct_0(k12_normform(A)),u1_struct_0(k12_normform(A))) )
=> ( B = k8_heyting1(A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(k12_normform(A)))
=> k8_funct_2(u1_struct_0(k12_normform(A)),u1_struct_0(k12_normform(A)),B,C) = k9_normform(A,k6_heyting1(A,k3_heyting1(A,C))) ) ) ) ).
fof(d9_heyting1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(u1_struct_0(k12_normform(A)),u1_struct_0(k12_normform(A))),u1_struct_0(k12_normform(A)))
& m2_relset_1(B,k2_zfmisc_1(u1_struct_0(k12_normform(A)),u1_struct_0(k12_normform(A))),u1_struct_0(k12_normform(A))) )
=> ( B = k9_heyting1(A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(k12_normform(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k12_normform(A)))
=> k2_binop_1(u1_struct_0(k12_normform(A)),u1_struct_0(k12_normform(A)),u1_struct_0(k12_normform(A)),B,C,D) = k9_normform(A,k7_heyting1(A,k3_heyting1(A,C),k3_heyting1(A,D))) ) ) ) ) ).
fof(d10_heyting1,axiom,
! [A,B] :
( m1_subset_1(B,u1_struct_0(k12_normform(A)))
=> k10_heyting1(A,B) = k1_zfmisc_1(B) ) ).
fof(d11_heyting1,axiom,
! [A,B] :
( m1_subset_1(B,u1_struct_0(k12_normform(A)))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k12_normform(A)),u1_struct_0(k12_normform(A)))
& m2_relset_1(C,u1_struct_0(k12_normform(A)),u1_struct_0(k12_normform(A))) )
=> ( C = k11_heyting1(A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(k12_normform(A)))
=> k8_funct_2(u1_struct_0(k12_normform(A)),u1_struct_0(k12_normform(A)),C,D) = k4_xboole_0(B,D) ) ) ) ) ).
fof(t27_heyting1,axiom,
! [A,B] :
( m1_subset_1(B,u1_struct_0(k12_normform(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k12_normform(A)))
=> r3_lattices(k12_normform(A),k8_funct_2(u1_struct_0(k12_normform(A)),u1_struct_0(k12_normform(A)),k11_heyting1(A,B),C),B) ) ) ).
fof(t28_heyting1,axiom,
! [A,B] :
( m1_subset_1(B,u1_struct_0(k12_normform(A)))
=> k4_lattices(k12_normform(A),B,k8_funct_2(u1_struct_0(k12_normform(A)),u1_struct_0(k12_normform(A)),k8_heyting1(A),B)) = k5_lattices(k12_normform(A)) ) ).
fof(t29_heyting1,axiom,
! [A,B] :
( m1_subset_1(B,u1_struct_0(k12_normform(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k12_normform(A)))
=> r3_lattices(k12_normform(A),k4_lattices(k12_normform(A),B,k2_binop_1(u1_struct_0(k12_normform(A)),u1_struct_0(k12_normform(A)),u1_struct_0(k12_normform(A)),k9_heyting1(A),B,C)),C) ) ) ).
fof(t30_heyting1,axiom,
! [A,B] :
( m2_subset_1(B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k12_normform(A)))
=> ( k10_normform(A,k3_heyting1(A,C),k2_heyting1(A,B)) = k1_xboole_0
=> r3_lattices(k12_normform(A),k8_funct_2(k7_normform(A),u1_struct_0(k12_normform(A)),k4_heyting1(A),B),k8_funct_2(u1_struct_0(k12_normform(A)),u1_struct_0(k12_normform(A)),k8_heyting1(A),C)) ) ) ) ).
fof(t31_heyting1,axiom,
! [A,B] :
( m2_subset_1(B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k12_normform(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k12_normform(A)))
=> ( ( ! [E] :
( m2_subset_1(E,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ( r2_hidden(E,C)
=> r2_hidden(k1_normform(k5_finsub_1(A),k5_finsub_1(A),E,B),k7_normform(A)) ) )
& r3_lattices(k12_normform(A),k4_lattices(k12_normform(A),C,k8_funct_2(k7_normform(A),u1_struct_0(k12_normform(A)),k4_heyting1(A),B)),D) )
=> r3_lattices(k12_normform(A),k8_funct_2(k7_normform(A),u1_struct_0(k12_normform(A)),k4_heyting1(A),B),k2_binop_1(u1_struct_0(k12_normform(A)),u1_struct_0(k12_normform(A)),u1_struct_0(k12_normform(A)),k9_heyting1(A),C,D)) ) ) ) ) ).
fof(t32_heyting1,axiom,
$true ).
fof(t33_heyting1,axiom,
! [A,B] :
( m1_subset_1(B,u1_struct_0(k12_normform(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k12_normform(A)))
=> k4_filter_0(k12_normform(A),B,C) = k2_lattice2(u1_struct_0(k12_normform(A)),k12_normform(A),k10_heyting1(A,B),k6_funcop_1(u1_struct_0(k12_normform(A)),u1_struct_0(k12_normform(A)),u1_lattices(k12_normform(A)),k8_heyting1(A),k7_funcop_1(u1_struct_0(k12_normform(A)),u1_struct_0(k12_normform(A)),k9_heyting1(A),k11_heyting1(A,B),C))) ) ) ).
fof(t34_heyting1,axiom,
! [A] : k6_lattices(k12_normform(A)) = k1_tarski(k4_tarski(k1_xboole_0,k1_xboole_0)) ).
fof(reflexivity_r1_heyting1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k5_finsub_1(A))
& m1_subset_1(C,k5_finsub_1(A)) )
=> r1_heyting1(A,B,B) ) ).
fof(redefinition_r1_heyting1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k5_finsub_1(A))
& m1_subset_1(C,k5_finsub_1(A)) )
=> ( r1_heyting1(A,B,C)
<=> r1_tarski(B,C) ) ) ).
fof(dt_k1_heyting1,axiom,
! [A] : ~ v1_xboole_0(k1_heyting1(A)) ).
fof(dt_k2_heyting1,axiom,
! [A,B] :
( m1_subset_1(B,k7_normform(A))
=> m2_subset_1(k2_heyting1(A,B),k5_finsub_1(k7_normform(A)),k8_normform(A)) ) ).
fof(redefinition_k2_heyting1,axiom,
! [A,B] :
( m1_subset_1(B,k7_normform(A))
=> k2_heyting1(A,B) = k1_tarski(B) ) ).
fof(dt_k3_heyting1,axiom,
! [A,B] :
( m1_subset_1(B,u1_struct_0(k12_normform(A)))
=> m2_subset_1(k3_heyting1(A,B),k5_finsub_1(k7_normform(A)),k8_normform(A)) ) ).
fof(dt_k4_heyting1,axiom,
! [A] :
( v1_funct_1(k4_heyting1(A))
& v1_funct_2(k4_heyting1(A),k7_normform(A),u1_struct_0(k12_normform(A)))
& m2_relset_1(k4_heyting1(A),k7_normform(A),u1_struct_0(k12_normform(A))) ) ).
fof(dt_k5_heyting1,axiom,
! [A] :
( v1_funct_1(k5_heyting1(A))
& v1_funct_2(k5_heyting1(A),k2_zfmisc_1(k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A))),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)))
& m2_relset_1(k5_heyting1(A),k2_zfmisc_1(k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A))),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A))) ) ).
fof(dt_k6_heyting1,axiom,
! [A,B] :
( m1_subset_1(B,k5_finsub_1(k7_normform(A)))
=> m1_subset_1(k6_heyting1(A,B),k5_finsub_1(k7_normform(A))) ) ).
fof(dt_k7_heyting1,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k5_finsub_1(k7_normform(A)))
& m1_subset_1(C,k5_finsub_1(k7_normform(A))) )
=> m1_subset_1(k7_heyting1(A,B,C),k5_finsub_1(k7_normform(A))) ) ).
fof(dt_k8_heyting1,axiom,
! [A] :
( v1_funct_1(k8_heyting1(A))
& v1_funct_2(k8_heyting1(A),u1_struct_0(k12_normform(A)),u1_struct_0(k12_normform(A)))
& m2_relset_1(k8_heyting1(A),u1_struct_0(k12_normform(A)),u1_struct_0(k12_normform(A))) ) ).
fof(dt_k9_heyting1,axiom,
! [A] :
( v1_funct_1(k9_heyting1(A))
& v1_funct_2(k9_heyting1(A),k2_zfmisc_1(u1_struct_0(k12_normform(A)),u1_struct_0(k12_normform(A))),u1_struct_0(k12_normform(A)))
& m2_relset_1(k9_heyting1(A),k2_zfmisc_1(u1_struct_0(k12_normform(A)),u1_struct_0(k12_normform(A))),u1_struct_0(k12_normform(A))) ) ).
fof(dt_k10_heyting1,axiom,
! [A,B] :
( m1_subset_1(B,u1_struct_0(k12_normform(A)))
=> m1_subset_1(k10_heyting1(A,B),k5_finsub_1(u1_struct_0(k12_normform(A)))) ) ).
fof(dt_k11_heyting1,axiom,
! [A,B] :
( m1_subset_1(B,u1_struct_0(k12_normform(A)))
=> ( v1_funct_1(k11_heyting1(A,B))
& v1_funct_2(k11_heyting1(A,B),u1_struct_0(k12_normform(A)),u1_struct_0(k12_normform(A)))
& m2_relset_1(k11_heyting1(A,B),u1_struct_0(k12_normform(A)),u1_struct_0(k12_normform(A))) ) ) ).
fof(d6_heyting1,axiom,
! [A,B] :
( m1_subset_1(B,k5_finsub_1(k7_normform(A)))
=> k6_heyting1(A,B) = k3_xboole_0(k7_normform(A),a_2_0_heyting1(A,B)) ) ).
fof(d7_heyting1,axiom,
! [A,B] :
( m1_subset_1(B,k5_finsub_1(k7_normform(A)))
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(k7_normform(A)))
=> k7_heyting1(A,B,C) = k3_xboole_0(k7_normform(A),a_3_2_heyting1(A,B,C)) ) ) ).
fof(t16_heyting1,axiom,
! [A,B] :
( m1_subset_1(B,k5_finsub_1(k7_normform(A)))
=> ! [C] :
( m2_subset_1(C,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ~ ( r2_hidden(C,k6_heyting1(A,B))
& ! [D] :
( m2_fraenkel(D,k7_normform(A),k1_heyting1(A),k1_fraenkel(k7_normform(A),k1_heyting1(A)))
=> ~ ( ! [E] :
( m2_subset_1(E,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ( r2_hidden(E,B)
=> r2_hidden(k8_funct_2(k7_normform(A),k1_heyting1(A),D,E),k5_setwiseo(A,k2_domain_1(k5_finsub_1(A),k5_finsub_1(A),E),k3_domain_1(k5_finsub_1(A),k5_finsub_1(A),E))) ) )
& C = k4_tarski(a_3_0_heyting1(A,B,D),a_3_1_heyting1(A,B,D)) ) ) ) ) ) ).
fof(fraenkel_a_2_0_heyting1,axiom,
! [A,B,C] :
( m1_subset_1(C,k5_finsub_1(k7_normform(B)))
=> ( r2_hidden(A,a_2_0_heyting1(B,C))
<=> ? [D] :
( m2_fraenkel(D,k7_normform(B),k1_heyting1(B),k1_fraenkel(k7_normform(B),k1_heyting1(B)))
& A = k4_tarski(a_3_0_heyting1(B,C,D),a_3_1_heyting1(B,C,D))
& ! [E] :
( m2_subset_1(E,k2_zfmisc_1(k5_finsub_1(B),k5_finsub_1(B)),k7_normform(B))
=> ( r2_hidden(E,C)
=> r2_hidden(k8_funct_2(k7_normform(B),k1_heyting1(B),D,E),k5_setwiseo(B,k2_domain_1(k5_finsub_1(B),k5_finsub_1(B),E),k3_domain_1(k5_finsub_1(B),k5_finsub_1(B),E))) ) ) ) ) ) ).
fof(fraenkel_a_3_0_heyting1,axiom,
! [A,B,C,D] :
( ( m1_subset_1(C,k5_finsub_1(k7_normform(B)))
& m2_fraenkel(D,k7_normform(B),k1_heyting1(B),k1_fraenkel(k7_normform(B),k1_heyting1(B))) )
=> ( r2_hidden(A,a_3_0_heyting1(B,C,D))
<=> ? [E] :
( m2_subset_1(E,k2_zfmisc_1(k5_finsub_1(B),k5_finsub_1(B)),k7_normform(B))
& A = k8_funct_2(k7_normform(B),k1_heyting1(B),D,E)
& r2_hidden(k8_funct_2(k7_normform(B),k1_heyting1(B),D,E),k3_domain_1(k5_finsub_1(B),k5_finsub_1(B),E))
& r2_hidden(E,C) ) ) ) ).
fof(fraenkel_a_3_1_heyting1,axiom,
! [A,B,C,D] :
( ( m1_subset_1(C,k5_finsub_1(k7_normform(B)))
& m2_fraenkel(D,k7_normform(B),k1_heyting1(B),k1_fraenkel(k7_normform(B),k1_heyting1(B))) )
=> ( r2_hidden(A,a_3_1_heyting1(B,C,D))
<=> ? [E] :
( m2_subset_1(E,k2_zfmisc_1(k5_finsub_1(B),k5_finsub_1(B)),k7_normform(B))
& A = k8_funct_2(k7_normform(B),k1_heyting1(B),D,E)
& r2_hidden(k8_funct_2(k7_normform(B),k1_heyting1(B),D,E),k2_domain_1(k5_finsub_1(B),k5_finsub_1(B),E))
& r2_hidden(E,C) ) ) ) ).
fof(fraenkel_a_3_2_heyting1,axiom,
! [A,B,C,D] :
( ( m1_subset_1(C,k5_finsub_1(k7_normform(B)))
& m1_subset_1(D,k5_finsub_1(k7_normform(B))) )
=> ( r2_hidden(A,a_3_2_heyting1(B,C,D))
<=> ? [E] :
( m2_fraenkel(E,k7_normform(B),k2_zfmisc_1(k5_finsub_1(B),k5_finsub_1(B)),k1_fraenkel(k7_normform(B),k2_zfmisc_1(k5_finsub_1(B),k5_finsub_1(B))))
& A = k6_normform(k7_normform(B),B,C,k6_funcop_1(k2_zfmisc_1(k5_finsub_1(B),k5_finsub_1(B)),k7_normform(B),k5_heyting1(B),E,k6_funct_3(k2_zfmisc_1(k5_finsub_1(B),k5_finsub_1(B)),k7_normform(B))))
& r1_tarski(k8_setwiseo(k7_normform(B),k2_zfmisc_1(k5_finsub_1(B),k5_finsub_1(B)),E,C),D) ) ) ) ).
%------------------------------------------------------------------------------