SET007 Axioms: SET007+206.ax


%------------------------------------------------------------------------------
% File     : SET007+206 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Topological Spaces and Continuous Functions
% 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    : pre_topc [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   96 (  30 unit)
%            Number of atoms       :  337 (  37 equality)
%            Maximal formula depth :   16 (   5 average)
%            Number of connectives :  262 (  21 ~  ;   0  |;  88  &)
%                                         (  14 <=>; 139 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   26 (   1 propositional; 0-3 arity)
%            Number of functors    :   20 (   1 constant; 0-4 arity)
%            Number of variables   :  155 (   0 singleton; 143 !;  12 ?)
%            Maximal term depth    :    4 (   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_pre_topc,axiom,(
    ? [A] :
      ( l1_pre_topc(A)
      & v1_pre_topc(A) ) )).

fof(rc2_pre_topc,axiom,(
    ? [A] :
      ( l1_pre_topc(A)
      & ~ v3_struct_0(A)
      & v1_pre_topc(A)
      & v2_pre_topc(A) ) )).

fof(fc1_pre_topc,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ( v1_xboole_0(k1_pre_topc(A))
        & v1_membered(k1_pre_topc(A))
        & v2_membered(k1_pre_topc(A))
        & v3_membered(k1_pre_topc(A))
        & v4_membered(k1_pre_topc(A))
        & v5_membered(k1_pre_topc(A)) ) ) )).

fof(fc2_pre_topc,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ~ v1_xboole_0(k2_pre_topc(A)) ) )).

fof(rc3_pre_topc,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ? [B] :
          ( m1_pre_topc(B,A)
          & v1_pre_topc(B) ) ) )).

fof(rc4_pre_topc,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A) )
     => ? [B] :
          ( m1_pre_topc(B,A)
          & ~ v3_struct_0(B)
          & v1_pre_topc(B) ) ) )).

fof(cc1_pre_topc,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_pre_topc(B,A)
         => v2_pre_topc(B) ) ) )).

fof(fc3_pre_topc,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( ~ v3_struct_0(k3_pre_topc(A,B))
        & v1_pre_topc(k3_pre_topc(A,B)) ) ) )).

fof(rc5_pre_topc,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ? [B] :
          ( m1_pre_topc(B,A)
          & v1_pre_topc(B)
          & v2_pre_topc(B) ) ) )).

fof(fc4_pre_topc,axiom,(
    ! [A,B] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( v1_pre_topc(k3_pre_topc(A,B))
        & v2_pre_topc(k3_pre_topc(A,B)) ) ) )).

fof(fc5_pre_topc,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => v4_pre_topc(k2_pre_topc(A),A) ) )).

fof(rc6_pre_topc,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
          & v4_pre_topc(B,A) ) ) )).

fof(rc7_pre_topc,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)
          & v4_pre_topc(B,A) ) ) )).

fof(d1_pre_topc,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ( v2_pre_topc(A)
      <=> ( r2_hidden(u1_struct_0(A),u1_pre_topc(A))
          & ! [B] :
              ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ( r1_tarski(B,u1_pre_topc(A))
               => r2_hidden(k5_setfam_1(u1_struct_0(A),B),u1_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)))
                 => ( ( r2_hidden(B,u1_pre_topc(A))
                      & r2_hidden(C,u1_pre_topc(A)) )
                   => r2_hidden(k5_subset_1(u1_struct_0(A),B,C),u1_pre_topc(A)) ) ) ) ) ) ) )).

fof(t1_pre_topc,axiom,(
    $true )).

fof(t2_pre_topc,axiom,(
    $true )).

fof(t3_pre_topc,axiom,(
    $true )).

fof(t4_pre_topc,axiom,(
    $true )).

fof(t5_pre_topc,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => r2_hidden(k1_xboole_0,u1_pre_topc(A)) ) )).

fof(d2_pre_topc,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => k1_pre_topc(A) = k1_xboole_0 ) )).

fof(d3_pre_topc,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => k2_pre_topc(A) = u1_struct_0(A) ) )).

fof(t6_pre_topc,axiom,(
    $true )).

fof(t7_pre_topc,axiom,(
    $true )).

fof(t8_pre_topc,axiom,(
    $true )).

fof(t9_pre_topc,axiom,(
    $true )).

fof(t10_pre_topc,axiom,(
    $true )).

fof(t11_pre_topc,axiom,(
    $true )).

fof(t12_pre_topc,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => k2_pre_topc(A) = u1_struct_0(A) ) )).

fof(t13_pre_topc,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => r2_hidden(B,k2_pre_topc(A)) ) ) )).

fof(t14_pre_topc,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => r1_tarski(B,k2_pre_topc(A)) ) ) )).

fof(t15_pre_topc,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k5_subset_1(u1_struct_0(A),B,k2_pre_topc(A)) = B ) ) )).

fof(t16_pre_topc,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( r1_tarski(B,k2_pre_topc(A))
         => m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) ) )).

fof(t17_pre_topc,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k3_subset_1(u1_struct_0(A),B) = k6_subset_1(u1_struct_0(A),k2_pre_topc(A),B) ) ) )).

fof(t18_pre_topc,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k4_subset_1(u1_struct_0(A),B,k3_subset_1(u1_struct_0(A),B)) = k2_pre_topc(A) ) ) )).

fof(t19_pre_topc,axiom,(
    $true )).

fof(t20_pre_topc,axiom,(
    $true )).

fof(t21_pre_topc,axiom,(
    $true )).

fof(t22_pre_topc,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k6_subset_1(u1_struct_0(A),k2_pre_topc(A),k6_subset_1(u1_struct_0(A),k2_pre_topc(A),B)) = B ) ) )).

fof(t23_pre_topc,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( ~ ( B != k2_pre_topc(A)
                & k6_subset_1(u1_struct_0(A),k2_pre_topc(A),B) = k1_xboole_0 )
            & ~ ( k6_subset_1(u1_struct_0(A),k2_pre_topc(A),B) != k1_xboole_0
                & B = k2_pre_topc(A) ) ) ) ) )).

fof(t24_pre_topc,axiom,(
    ! [A] :
      ( l1_struct_0(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)))
             => ( k6_subset_1(u1_struct_0(A),k2_pre_topc(A),B) = C
               => k2_pre_topc(A) = k4_subset_1(u1_struct_0(A),B,C) ) ) ) ) )).

fof(t25_pre_topc,axiom,(
    ! [A] :
      ( l1_struct_0(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)))
             => ( ( k2_pre_topc(A) = k4_subset_1(u1_struct_0(A),B,C)
                  & r1_xboole_0(B,C) )
               => C = k6_subset_1(u1_struct_0(A),k2_pre_topc(A),B) ) ) ) ) )).

fof(t26_pre_topc,axiom,(
    $true )).

fof(t27_pre_topc,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => k2_pre_topc(A) = k3_subset_1(u1_struct_0(A),k1_pre_topc(A)) ) )).

fof(d4_pre_topc,axiom,(
    $true )).

fof(d5_pre_topc,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v3_pre_topc(B,A)
          <=> r2_hidden(B,u1_pre_topc(A)) ) ) ) )).

fof(d6_pre_topc,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v4_pre_topc(B,A)
          <=> v3_pre_topc(k6_subset_1(u1_struct_0(A),k2_pre_topc(A),B),A) ) ) ) )).

fof(d7_pre_topc,axiom,(
    $true )).

fof(d8_pre_topc,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( r1_pre_topc(A,B)
          <=> k2_pre_topc(A) = k5_setfam_1(u1_struct_0(A),B) ) ) ) )).

fof(d9_pre_topc,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( l1_pre_topc(B)
         => ( m1_pre_topc(B,A)
          <=> ( r1_tarski(k2_pre_topc(B),k2_pre_topc(A))
              & ! [C] :
                  ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
                 => ( r2_hidden(C,u1_pre_topc(B))
                  <=> ? [D] :
                        ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                        & r2_hidden(D,u1_pre_topc(A))
                        & C = k3_xboole_0(D,k2_pre_topc(B)) ) ) ) ) ) ) ) )).

fof(d10_pre_topc,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( ( v1_pre_topc(C)
                & m1_pre_topc(C,A) )
             => ( C = k3_pre_topc(A,B)
              <=> k2_pre_topc(C) = B ) ) ) ) )).

fof(d11_pre_topc,axiom,(
    $true )).

fof(d12_pre_topc,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( l1_pre_topc(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( v5_pre_topc(C,A,B)
              <=> ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
                   => ( v4_pre_topc(D,B)
                     => v4_pre_topc(k5_pre_topc(A,B,C,D),A) ) ) ) ) ) ) )).

fof(t28_pre_topc,axiom,(
    $true )).

fof(t29_pre_topc,axiom,(
    $true )).

fof(t30_pre_topc,axiom,(
    $true )).

fof(t31_pre_topc,axiom,(
    $true )).

fof(t32_pre_topc,axiom,(
    $true )).

fof(t33_pre_topc,axiom,(
    $true )).

fof(t34_pre_topc,axiom,(
    $true )).

fof(t35_pre_topc,axiom,(
    $true )).

fof(t36_pre_topc,axiom,(
    $true )).

fof(t37_pre_topc,axiom,(
    $true )).

fof(t38_pre_topc,axiom,(
    $true )).

fof(t39_pre_topc,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_pre_topc(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
             => m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) ) ) ) )).

fof(t40_pre_topc,axiom,(
    $true )).

fof(t41_pre_topc,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ~ ( B != k1_pre_topc(A)
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ~ r2_hidden(C,B) ) ) ) ) )).

fof(t42_pre_topc,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => v4_pre_topc(k2_pre_topc(A),A) ) )).

fof(t43_pre_topc,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_pre_topc(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
             => ( v4_pre_topc(C,B)
              <=> ? [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                    & v4_pre_topc(D,A)
                    & k3_xboole_0(D,k2_pre_topc(B)) = C ) ) ) ) ) )).

fof(t44_pre_topc,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
               => ( r2_hidden(C,B)
                 => v4_pre_topc(C,A) ) )
           => v4_pre_topc(k6_setfam_1(u1_struct_0(A),B),A) ) ) ) )).

fof(d13_pre_topc,axiom,(
    ! [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)))
             => ( C = k6_pre_topc(A,B)
              <=> ! [D] :
                    ( r2_hidden(D,u1_struct_0(A))
                   => ( r2_hidden(D,C)
                    <=> ! [E] :
                          ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                         => ~ ( v3_pre_topc(E,A)
                              & r2_hidden(D,E)
                              & r1_xboole_0(B,E) ) ) ) ) ) ) ) ) )).

fof(t45_pre_topc,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( r2_hidden(C,u1_struct_0(A))
             => ( r2_hidden(C,k6_pre_topc(A,B))
              <=> ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                   => ( ( v4_pre_topc(D,A)
                        & r1_tarski(B,D) )
                     => r2_hidden(C,D) ) ) ) ) ) ) )).

fof(t46_pre_topc,axiom,(
    ! [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(k1_zfmisc_1(u1_struct_0(A))))
              & ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => ( r2_hidden(D,C)
                  <=> ( v4_pre_topc(D,A)
                      & r1_tarski(B,D) ) ) )
              & k6_pre_topc(A,B) = k6_setfam_1(u1_struct_0(A),C) ) ) ) )).

fof(t47_pre_topc,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_pre_topc(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
                 => ( C = D
                   => k6_pre_topc(B,D) = k3_xboole_0(k6_pre_topc(A,C),k2_pre_topc(B)) ) ) ) ) ) )).

fof(t48_pre_topc,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => r1_tarski(B,k6_pre_topc(A,B)) ) ) )).

fof(t49_pre_topc,axiom,(
    ! [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)))
             => ( r1_tarski(B,C)
               => r1_tarski(k6_pre_topc(A,B),k6_pre_topc(A,C)) ) ) ) ) )).

fof(t50_pre_topc,axiom,(
    ! [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)))
             => k6_pre_topc(A,k4_subset_1(u1_struct_0(A),B,C)) = k4_subset_1(u1_struct_0(A),k6_pre_topc(A,B),k6_pre_topc(A,C)) ) ) ) )).

fof(t51_pre_topc,axiom,(
    ! [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)))
             => r1_tarski(k6_pre_topc(A,k5_subset_1(u1_struct_0(A),B,C)),k5_subset_1(u1_struct_0(A),k6_pre_topc(A,B),k6_pre_topc(A,C))) ) ) ) )).

fof(t52_pre_topc,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( ( v4_pre_topc(B,A)
             => k6_pre_topc(A,B) = B )
            & ( ( v2_pre_topc(A)
                & k6_pre_topc(A,B) = B )
             => v4_pre_topc(B,A) ) ) ) ) )).

fof(t53_pre_topc,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( ( v3_pre_topc(B,A)
             => k6_pre_topc(A,k6_subset_1(u1_struct_0(A),k2_pre_topc(A),B)) = k6_subset_1(u1_struct_0(A),k2_pre_topc(A),B) )
            & ( ( v2_pre_topc(A)
                & k6_pre_topc(A,k6_subset_1(u1_struct_0(A),k2_pre_topc(A),B)) = k6_subset_1(u1_struct_0(A),k2_pre_topc(A),B) )
             => v3_pre_topc(B,A) ) ) ) ) )).

fof(t54_pre_topc,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r2_hidden(C,k6_pre_topc(A,B))
              <=> ( ~ v3_struct_0(A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                     => ~ ( v3_pre_topc(D,A)
                          & r2_hidden(C,D)
                          & r1_xboole_0(B,D) ) ) ) ) ) ) ) )).

fof(dt_m1_pre_topc,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_pre_topc(B,A)
         => l1_pre_topc(B) ) ) )).

fof(existence_m1_pre_topc,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ? [B] : m1_pre_topc(B,A) ) )).

fof(dt_l1_pre_topc,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => l1_struct_0(A) ) )).

fof(existence_l1_pre_topc,axiom,(
    ? [A] : l1_pre_topc(A) )).

fof(abstractness_v1_pre_topc,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ( v1_pre_topc(A)
       => A = g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) ) ) )).

fof(dt_k1_pre_topc,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => m1_subset_1(k1_pre_topc(A),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k2_pre_topc,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => m1_subset_1(k2_pre_topc(A),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k3_pre_topc,axiom,(
    ! [A,B] :
      ( ( l1_pre_topc(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( v1_pre_topc(k3_pre_topc(A,B))
        & m1_pre_topc(k3_pre_topc(A,B),A) ) ) )).

fof(dt_k4_pre_topc,axiom,(
    ! [A,B,C,D] :
      ( ( l1_struct_0(A)
        & l1_struct_0(B)
        & v1_funct_1(C)
        & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
        & m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
     => m1_subset_1(k4_pre_topc(A,B,C,D),k1_zfmisc_1(u1_struct_0(B))) ) )).

fof(redefinition_k4_pre_topc,axiom,(
    ! [A,B,C,D] :
      ( ( l1_struct_0(A)
        & l1_struct_0(B)
        & v1_funct_1(C)
        & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
        & m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
     => k4_pre_topc(A,B,C,D) = k9_relat_1(C,D) ) )).

fof(dt_k5_pre_topc,axiom,(
    ! [A,B,C,D] :
      ( ( l1_struct_0(A)
        & l1_struct_0(B)
        & v1_funct_1(C)
        & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
        & m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
     => m1_subset_1(k5_pre_topc(A,B,C,D),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(redefinition_k5_pre_topc,axiom,(
    ! [A,B,C,D] :
      ( ( l1_struct_0(A)
        & l1_struct_0(B)
        & v1_funct_1(C)
        & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
        & m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
     => k5_pre_topc(A,B,C,D) = k10_relat_1(C,D) ) )).

fof(dt_k6_pre_topc,axiom,(
    ! [A,B] :
      ( ( l1_pre_topc(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => m1_subset_1(k6_pre_topc(A,B),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_u1_pre_topc,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => m1_subset_1(u1_pre_topc(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) )).

fof(dt_g1_pre_topc,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( v1_pre_topc(g1_pre_topc(A,B))
        & l1_pre_topc(g1_pre_topc(A,B)) ) ) )).

fof(free_g1_pre_topc,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ! [C,D] :
          ( g1_pre_topc(A,B) = g1_pre_topc(C,D)
         => ( A = C
            & B = D ) ) ) )).
%------------------------------------------------------------------------------