SET007 Axioms: SET007+291.ax


%------------------------------------------------------------------------------
% File     : SET007+291 : TPTP v7.5.0. 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 unit)
%            Number of atoms       :  246 (  42 equality)
%            Maximal formula depth :   19 (   6 average)
%            Number of connectives :  201 (  20 ~  ;   0  |;  68  &)
%                                         (  11 <=>; 102 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   27 (   1 propositional; 0-4 arity)
%            Number of functors    :   50 (   1 constant; 0-6 arity)
%            Number of variables   :  159 (   0 singleton; 154 !;   5 ?)
%            Maximal term depth    :    6 (   2 average)
% 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) ) ) ) )).
%------------------------------------------------------------------------------