SET007 Axioms: SET007+371.ax


%------------------------------------------------------------------------------
% File     : SET007+371 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : On Discrete and Almost Discrete Topological Spaces
% 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    : tex_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   77 (   6 unit)
%            Number of atoms       :  499 (  64 equality)
%            Maximal formula depth :   14 (   7 average)
%            Number of connectives :  564 ( 142 ~  ;   7  |; 238  &)
%                                         (  13 <=>; 164 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   24 (   1 propositional; 0-2 arity)
%            Number of functors    :   22 (   1 constant; 0-3 arity)
%            Number of variables   :  150 (   0 singleton; 134 !;  16 ?)
%            Maximal term depth    :    6 (   1 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(fc1_tex_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( v1_struct_0(g1_struct_0(A))
        & ~ v3_struct_0(g1_struct_0(A)) ) ) )).

fof(rc1_tex_1,axiom,(
    ? [A] :
      ( l1_struct_0(A)
      & v1_struct_0(A)
      & ~ v3_struct_0(A)
      & v3_realset2(A) ) )).

fof(rc2_tex_1,axiom,(
    ? [A] :
      ( l1_struct_0(A)
      & v1_struct_0(A)
      & ~ v3_struct_0(A)
      & ~ v3_realset2(A) ) )).

fof(rc3_tex_1,axiom,(
    ? [A] :
      ( l1_pre_topc(A)
      & ~ v3_struct_0(A)
      & v3_realset2(A)
      & v1_pre_topc(A) ) )).

fof(rc4_tex_1,axiom,(
    ? [A] :
      ( l1_pre_topc(A)
      & ~ v3_struct_0(A)
      & ~ v3_realset2(A)
      & v1_pre_topc(A) ) )).

fof(rc5_tex_1,axiom,(
    ? [A] :
      ( l1_pre_topc(A)
      & ~ v3_struct_0(A)
      & v3_realset2(A)
      & v1_pre_topc(A)
      & v2_pre_topc(A) ) )).

fof(cc1_tex_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ( ( ~ v3_struct_0(A)
          & v3_realset2(A)
          & v2_pre_topc(A) )
       => ( ~ v3_struct_0(A)
          & v2_pre_topc(A)
          & v1_tdlat_3(A)
          & v2_tdlat_3(A)
          & v3_tdlat_3(A)
          & v4_tdlat_3(A)
          & v5_tdlat_3(A) ) ) ) )).

fof(cc2_tex_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ( ( ~ v3_struct_0(A)
          & v2_pre_topc(A)
          & v1_tdlat_3(A)
          & v2_tdlat_3(A) )
       => ( ~ v3_struct_0(A)
          & v3_realset2(A)
          & v2_pre_topc(A) ) ) ) )).

fof(rc6_tex_1,axiom,(
    ? [A] :
      ( l1_pre_topc(A)
      & ~ v3_struct_0(A)
      & ~ v3_realset2(A)
      & v1_pre_topc(A)
      & v2_pre_topc(A) ) )).

fof(cc3_tex_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ( ( ~ v3_struct_0(A)
          & v2_pre_topc(A)
          & ~ v1_tdlat_3(A) )
       => ( ~ v3_struct_0(A)
          & ~ v3_realset2(A)
          & v2_pre_topc(A) ) ) ) )).

fof(cc4_tex_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ( ( ~ v3_struct_0(A)
          & v2_pre_topc(A)
          & ~ v2_tdlat_3(A) )
       => ( ~ v3_struct_0(A)
          & ~ v3_realset2(A)
          & v2_pre_topc(A) ) ) ) )).

fof(fc2_tex_1,axiom,(
    ! [A] : ~ v1_xboole_0(k1_tex_1(A)) )).

fof(fc3_tex_1,axiom,(
    ! [A] :
      ( v1_pre_topc(k2_tex_1(A))
      & v2_pre_topc(k2_tex_1(A))
      & v2_tdlat_3(k2_tex_1(A))
      & v3_tdlat_3(k2_tex_1(A)) ) )).

fof(fc4_tex_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ~ v3_struct_0(k2_tex_1(A))
        & v1_pre_topc(k2_tex_1(A))
        & v2_pre_topc(k2_tex_1(A))
        & v2_tdlat_3(k2_tex_1(A))
        & v3_tdlat_3(k2_tex_1(A))
        & v4_tdlat_3(k2_tex_1(A))
        & v5_tdlat_3(k2_tex_1(A)) ) ) )).

fof(fc5_tex_1,axiom,(
    ! [A] :
      ( v1_pre_topc(k2_pcomps_1(A))
      & v2_pre_topc(k2_pcomps_1(A))
      & v1_tdlat_3(k2_pcomps_1(A))
      & v3_tdlat_3(k2_pcomps_1(A)) ) )).

fof(rc7_tex_1,axiom,(
    ? [A] :
      ( l1_pre_topc(A)
      & ~ v3_struct_0(A)
      & ~ v3_realset2(A)
      & v1_pre_topc(A)
      & v2_pre_topc(A)
      & v1_tdlat_3(A)
      & ~ v2_tdlat_3(A)
      & v3_tdlat_3(A)
      & v4_tdlat_3(A)
      & v5_tdlat_3(A) ) )).

fof(rc8_tex_1,axiom,(
    ? [A] :
      ( l1_pre_topc(A)
      & ~ v3_struct_0(A)
      & ~ v3_realset2(A)
      & v1_pre_topc(A)
      & v2_pre_topc(A)
      & ~ v1_tdlat_3(A)
      & v2_tdlat_3(A)
      & v3_tdlat_3(A)
      & v4_tdlat_3(A)
      & v5_tdlat_3(A) ) )).

fof(fc6_tex_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,A)
     => ( v1_pre_topc(k4_tex_1(A,B))
        & v2_pre_topc(k4_tex_1(A,B)) ) ) )).

fof(fc7_tex_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A) )
     => ( ~ v3_struct_0(k4_tex_1(A,B))
        & v1_pre_topc(k4_tex_1(A,B))
        & v2_pre_topc(k4_tex_1(A,B)) ) ) )).

fof(rc9_tex_1,axiom,(
    ? [A] :
      ( l1_pre_topc(A)
      & ~ v3_struct_0(A)
      & ~ v3_realset2(A)
      & v1_pre_topc(A)
      & v2_pre_topc(A)
      & ~ v1_tdlat_3(A)
      & ~ v2_tdlat_3(A) ) )).

fof(cc5_tex_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ( ( ~ v3_struct_0(A)
          & v2_pre_topc(A)
          & ~ v3_tdlat_3(A) )
       => ( ~ v3_struct_0(A)
          & ~ v3_realset2(A)
          & v2_pre_topc(A)
          & ~ v1_tdlat_3(A)
          & ~ v2_tdlat_3(A) ) ) ) )).

fof(rc10_tex_1,axiom,(
    ? [A] :
      ( l1_pre_topc(A)
      & ~ v3_struct_0(A)
      & ~ v3_realset2(A)
      & v1_pre_topc(A)
      & v2_pre_topc(A)
      & ~ v1_tdlat_3(A)
      & ~ v2_tdlat_3(A)
      & v3_tdlat_3(A)
      & v4_tdlat_3(A)
      & v5_tdlat_3(A) ) )).

fof(rc11_tex_1,axiom,(
    ? [A] :
      ( l1_pre_topc(A)
      & ~ v3_struct_0(A)
      & ~ v3_realset2(A)
      & v1_pre_topc(A)
      & v2_pre_topc(A)
      & ~ v1_tdlat_3(A)
      & ~ v2_tdlat_3(A)
      & ~ v3_tdlat_3(A) ) )).

fof(t1_tex_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k6_tmap_1(A,B))))
                 => ( ( C = D
                      & v3_pre_topc(C,A) )
                   => v3_pre_topc(D,k6_tmap_1(A,B)) ) ) ) ) ) )).

fof(t2_tex_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k6_tmap_1(A,B))))
                 => ( ( C = D
                      & v4_pre_topc(C,A) )
                   => v4_pre_topc(D,k6_tmap_1(A,B)) ) ) ) ) ) )).

fof(t3_tex_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_tmap_1(A,k3_subset_1(u1_struct_0(A),B)))))
             => ( C = B
               => v4_pre_topc(C,k6_tmap_1(A,k3_subset_1(u1_struct_0(A),B))) ) ) ) ) )).

fof(t4_tex_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_tmap_1(A,B))))
             => ( ( C = B
                  & v1_tops_1(B,A) )
               => ( v1_tops_1(C,k6_tmap_1(A,B))
                  & v3_pre_topc(C,k6_tmap_1(A,B)) ) ) ) ) ) )).

fof(t5_tex_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_tmap_1(A,B))))
             => ( ( r1_tarski(B,C)
                  & v1_tops_1(B,A) )
               => v1_tops_3(C,k6_tmap_1(A,B)) ) ) ) ) )).

fof(t6_tex_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_tmap_1(A,k3_subset_1(u1_struct_0(A),B)))))
             => ( ( C = B
                  & v2_tops_1(B,A) )
               => ( v2_tops_1(C,k6_tmap_1(A,k3_subset_1(u1_struct_0(A),B)))
                  & v4_pre_topc(C,k6_tmap_1(A,k3_subset_1(u1_struct_0(A),B))) ) ) ) ) ) )).

fof(t7_tex_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_tmap_1(A,k3_subset_1(u1_struct_0(A),B)))))
             => ( ( r1_tarski(C,B)
                  & v2_tops_1(B,A) )
               => v3_tops_1(C,k6_tmap_1(A,k3_subset_1(u1_struct_0(A),B))) ) ) ) ) )).

fof(d1_tex_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ( v3_realset2(A)
      <=> ? [B] :
            ( m1_subset_1(B,u1_struct_0(A))
            & u1_struct_0(A) = k1_struct_0(A,B) ) ) ) )).

fof(t8_tex_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_realset2(A)
        & l1_pre_topc(A) )
     => ( v3_tdlat_3(A)
       => ( v1_xboole_0(u1_pre_topc(A))
          | v2_pre_topc(A) ) ) ) )).

fof(d2_tex_1,axiom,(
    ! [A] : k1_tex_1(A) = k2_tarski(k1_xboole_0,A) )).

fof(d3_tex_1,axiom,(
    ! [A] : k2_tex_1(A) = g1_pre_topc(A,k1_tex_1(A)) )).

fof(t9_tex_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v2_tdlat_3(A)
        & l1_pre_topc(A) )
     => g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) = k2_tex_1(u1_struct_0(A)) ) )).

fof(t10_tex_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) = g1_pre_topc(u1_struct_0(k2_tex_1(u1_struct_0(A))),u1_pre_topc(k2_tex_1(u1_struct_0(A))))
       => v2_tdlat_3(A) ) ) )).

fof(t11_tex_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v2_tdlat_3(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( ( v1_xboole_0(B)
             => k6_pre_topc(A,B) = k1_xboole_0 )
            & ( ~ v1_xboole_0(B)
             => k6_pre_topc(A,B) = u1_struct_0(A) ) ) ) ) )).

fof(t12_tex_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v2_tdlat_3(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( ( B != u1_struct_0(A)
             => k1_tops_1(A,B) = k1_xboole_0 )
            & ( B = u1_struct_0(A)
             => k1_tops_1(A,B) = u1_struct_0(A) ) ) ) ) )).

fof(t13_tex_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
           => ( ~ v1_xboole_0(B)
             => k6_pre_topc(A,B) = u1_struct_0(A) ) )
       => v2_tdlat_3(A) ) ) )).

fof(t14_tex_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
           => ( B != u1_struct_0(A)
             => k1_tops_1(A,B) = k1_xboole_0 ) )
       => v2_tdlat_3(A) ) ) )).

fof(t15_tex_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v2_tdlat_3(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( ( B != k1_xboole_0
             => v1_tops_1(B,A) )
            & ( B != u1_struct_0(A)
             => v2_tops_1(B,A) ) ) ) ) )).

fof(t16_tex_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
           => ( B != k1_xboole_0
             => v1_tops_1(B,A) ) )
       => v2_tdlat_3(A) ) ) )).

fof(t17_tex_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
           => ( B != u1_struct_0(A)
             => v2_tops_1(B,A) ) )
       => v2_tdlat_3(A) ) ) )).

fof(t18_tex_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v1_tdlat_3(A)
        & l1_pre_topc(A) )
     => g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) = k2_pcomps_1(u1_struct_0(A)) ) )).

fof(t19_tex_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) = g1_pre_topc(u1_struct_0(k2_pcomps_1(u1_struct_0(A))),u1_pre_topc(k2_pcomps_1(u1_struct_0(A))))
       => v1_tdlat_3(A) ) ) )).

fof(t20_tex_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v1_tdlat_3(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( k6_pre_topc(A,B) = B
            & k1_tops_1(A,B) = B ) ) ) )).

fof(t21_tex_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
           => k6_pre_topc(A,B) = B )
       => v1_tdlat_3(A) ) ) )).

fof(t22_tex_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
           => k1_tops_1(A,B) = B )
       => v1_tdlat_3(A) ) ) )).

fof(t23_tex_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( k2_tex_1(A) = k2_pcomps_1(A)
      <=> ? [B] :
            ( m1_subset_1(B,A)
            & A = k1_tarski(B) ) ) ) )).

fof(d4_tex_1,axiom,(
    $true )).

fof(t24_tex_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k4_tex_1(A,B))))
             => ( ( r1_tarski(k1_tarski(B),C)
                 => v4_pre_topc(C,k4_tex_1(A,B)) )
                & ( v4_pre_topc(C,k4_tex_1(A,B))
                 => ( v1_xboole_0(C)
                    | r1_tarski(k1_tarski(B),C) ) ) ) ) ) ) )).

fof(t25_tex_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ( ~ v1_xboole_0(k4_xboole_0(A,k1_tarski(B)))
           => ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k4_tex_1(A,B))))
               => ( ( C = k1_tarski(B)
                   => ( v4_pre_topc(C,k4_tex_1(A,B))
                      & v2_tops_1(C,k4_tex_1(A,B)) ) )
                  & ( ( v4_pre_topc(C,k4_tex_1(A,B))
                      & v2_tops_1(C,k4_tex_1(A,B)) )
                   => ( v1_xboole_0(C)
                      | C = k1_tarski(B) ) ) ) ) ) ) ) )).

fof(t26_tex_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k4_tex_1(A,B))))
             => ( ( r1_tarski(C,k4_xboole_0(A,k1_tarski(B)))
                 => v3_pre_topc(C,k4_tex_1(A,B)) )
                & ( v3_pre_topc(C,k4_tex_1(A,B))
                 => ( C = A
                    | r1_tarski(C,k4_xboole_0(A,k1_tarski(B))) ) ) ) ) ) ) )).

fof(t27_tex_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ( ~ v1_xboole_0(k4_xboole_0(A,k1_tarski(B)))
           => ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k4_tex_1(A,B))))
               => ( ( C = k4_xboole_0(A,k1_tarski(B))
                   => ( v3_pre_topc(C,k4_tex_1(A,B))
                      & v1_tops_1(C,k4_tex_1(A,B)) ) )
                  & ( ( v3_pre_topc(C,k4_tex_1(A,B))
                      & v1_tops_1(C,k4_tex_1(A,B)) )
                   => ( C = A
                      | C = k4_xboole_0(A,k1_tarski(B)) ) ) ) ) ) ) ) )).

fof(t28_tex_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_pre_topc(C)
                & l1_pre_topc(C) )
             => ( g1_pre_topc(u1_struct_0(C),u1_pre_topc(C)) = g1_pre_topc(u1_struct_0(k4_tex_1(A,B)),u1_pre_topc(k4_tex_1(A,B)))
              <=> ( u1_struct_0(C) = A
                  & ! [D] :
                      ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(C)))
                     => ( ( r1_tarski(k1_tarski(B),D)
                         => v4_pre_topc(D,C) )
                        & ( v4_pre_topc(D,C)
                         => ( v1_xboole_0(D)
                            | r1_tarski(k1_tarski(B),D) ) ) ) ) ) ) ) ) ) )).

fof(t29_tex_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_pre_topc(C)
                & l1_pre_topc(C) )
             => ( g1_pre_topc(u1_struct_0(C),u1_pre_topc(C)) = g1_pre_topc(u1_struct_0(k4_tex_1(A,B)),u1_pre_topc(k4_tex_1(A,B)))
              <=> ( u1_struct_0(C) = A
                  & ! [D] :
                      ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(C)))
                     => ( ( r1_tarski(D,k4_xboole_0(A,k1_tarski(B)))
                         => v3_pre_topc(D,C) )
                        & ( v3_pre_topc(D,C)
                         => ( D = A
                            | r1_tarski(D,k4_xboole_0(A,k1_tarski(B))) ) ) ) ) ) ) ) ) ) )).

fof(t30_tex_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_pre_topc(C)
                & l1_pre_topc(C) )
             => ( g1_pre_topc(u1_struct_0(C),u1_pre_topc(C)) = g1_pre_topc(u1_struct_0(k4_tex_1(A,B)),u1_pre_topc(k4_tex_1(A,B)))
              <=> ( u1_struct_0(C) = A
                  & ! [D] :
                      ( ( ~ v1_xboole_0(D)
                        & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(C))) )
                     => k6_pre_topc(C,D) = k2_xboole_0(D,k1_tarski(B)) ) ) ) ) ) ) )).

fof(t31_tex_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_pre_topc(C)
                & l1_pre_topc(C) )
             => ( g1_pre_topc(u1_struct_0(C),u1_pre_topc(C)) = g1_pre_topc(u1_struct_0(k4_tex_1(A,B)),u1_pre_topc(k4_tex_1(A,B)))
              <=> ( u1_struct_0(C) = A
                  & ! [D] :
                      ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(C)))
                     => ( D != A
                       => k1_tops_1(C,D) = k4_xboole_0(D,k1_tarski(B)) ) ) ) ) ) ) ) )).

fof(t32_tex_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ( k4_tex_1(A,B) = k2_tex_1(A)
          <=> A = k1_tarski(B) ) ) ) )).

fof(t33_tex_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ( k4_tex_1(A,B) = k2_pcomps_1(A)
          <=> A = k1_tarski(B) ) ) ) )).

fof(t34_tex_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k4_tex_1(A,B))))
             => ( C = k1_tarski(B)
               => k2_pcomps_1(A) = k6_tmap_1(k4_tex_1(A,B),C) ) ) ) ) )).

fof(d6_tex_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v1_tdlat_3(A)
      <=> ! [B] :
            ( ( ~ v1_xboole_0(B)
              & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
           => ~ v2_tops_1(B,A) ) ) ) )).

fof(t35_tex_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v1_tdlat_3(A)
      <=> ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
           => ~ ( B != u1_struct_0(A)
                & v1_tops_1(B,A) ) ) ) ) )).

fof(d7_tex_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v3_tdlat_3(A)
      <=> ! [B] :
            ( ( ~ v1_xboole_0(B)
              & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
           => ~ v3_tops_1(B,A) ) ) ) )).

fof(t36_tex_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v3_tdlat_3(A)
      <=> ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
           => ~ ( B != u1_struct_0(A)
                & v1_tops_3(B,A) ) ) ) ) )).

fof(t37_tex_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( ~ ( ~ v3_tdlat_3(A)
            & ! [B] :
                ( ( ~ v1_xboole_0(B)
                  & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
               => ~ ( v2_tops_1(B,A)
                    & v4_pre_topc(B,A) ) ) )
        & ~ ( ? [B] :
                ( ~ v1_xboole_0(B)
                & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
                & v2_tops_1(B,A)
                & v4_pre_topc(B,A) )
            & v3_tdlat_3(A) ) ) ) )).

fof(t38_tex_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( ~ ( ~ v3_tdlat_3(A)
            & ! [B] :
                ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
               => ~ ( B != u1_struct_0(A)
                    & v1_tops_1(B,A)
                    & v3_pre_topc(B,A) ) ) )
        & ~ ( ? [B] :
                ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
                & B != u1_struct_0(A)
                & v1_tops_1(B,A)
                & v3_pre_topc(B,A) )
            & v3_tdlat_3(A) ) ) ) )).

fof(t39_tex_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ( ~ ( ~ v1_xboole_0(k4_xboole_0(A,k1_tarski(B)))
                & v3_tdlat_3(k4_tex_1(A,B)) )
            & ~ ( ~ v3_tdlat_3(k4_tex_1(A,B))
                & v1_xboole_0(k4_xboole_0(A,k1_tarski(B))) ) ) ) ) )).

fof(t40_tex_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ~ ( v2_tops_1(B,A)
              & v3_tdlat_3(k6_tmap_1(A,k3_subset_1(u1_struct_0(A),B))) ) ) ) )).

fof(t41_tex_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ~ ( B != u1_struct_0(A)
              & v1_tops_1(B,A)
              & v3_tdlat_3(k6_tmap_1(A,B)) ) ) ) )).

fof(dt_k1_tex_1,axiom,(
    ! [A] : m1_subset_1(k1_tex_1(A),k1_zfmisc_1(k1_zfmisc_1(A))) )).

fof(dt_k2_tex_1,axiom,(
    ! [A] : l1_pre_topc(k2_tex_1(A)) )).

fof(dt_k3_tex_1,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => m1_subset_1(k3_tex_1(A,B,C),k1_zfmisc_1(k1_zfmisc_1(A))) ) )).

fof(redefinition_k3_tex_1,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => k3_tex_1(A,B,C) = k4_xboole_0(B,C) ) )).

fof(dt_k4_tex_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,A)
     => l1_pre_topc(k4_tex_1(A,B)) ) )).

fof(d5_tex_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,A)
     => k4_tex_1(A,B) = g1_pre_topc(A,k3_tex_1(A,k1_pcomps_1(A),a_2_0_tex_1(A,B))) ) )).

fof(fraenkel_a_2_0_tex_1,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,B)
     => ( r2_hidden(A,a_2_0_tex_1(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,k1_zfmisc_1(B))
            & A = D
            & r2_hidden(C,D)
            & D != B ) ) ) )).
%------------------------------------------------------------------------------