SET007 Axioms: SET007+882.ax


%------------------------------------------------------------------------------
% File     : SET007+882 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : On the characteristic and weight of a topological space
% 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    : topgen_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   70 (   3 unit)
%            Number of atoms       :  403 (  45 equality)
%            Maximal formula depth :   20 (   8 average)
%            Number of connectives :  402 (  69 ~  ;   1  |; 168  &)
%                                         (  18 <=>; 146 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   30 (   1 propositional; 0-3 arity)
%            Number of functors    :   46 (   1 constant; 0-4 arity)
%            Number of variables   :  202 (   0 singleton; 181 !;  21 ?)
%            Maximal term depth    :    5 (   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_topgen_2,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))))
          & ~ v1_xboole_0(B)
          & v1_tops_2(B,A) ) ) )).

fof(cc1_topgen_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ( v6_group_1(A)
       => v1_topgen_2(A) ) ) )).

fof(cc2_topgen_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ( ~ v1_topgen_2(A)
       => ( ~ v3_struct_0(A)
          & ~ v6_group_1(A)
          & ~ v3_realset2(A) ) ) ) )).

fof(rc2_topgen_2,axiom,(
    ? [A] :
      ( l1_pre_topc(A)
      & ~ v3_struct_0(A)
      & v2_pre_topc(A)
      & v6_group_1(A)
      & v1_topgen_2(A) ) )).

fof(rc3_topgen_2,axiom,(
    ? [A] :
      ( l1_pre_topc(A)
      & ~ v3_struct_0(A)
      & v2_pre_topc(A)
      & ~ v6_group_1(A)
      & ~ v3_realset2(A)
      & ~ v1_topgen_2(A) ) )).

fof(fc1_topgen_2,axiom,(
    ! [A,B] :
      ( v1_pre_topc(k5_topgen_2(A,B))
      & v2_pre_topc(k5_topgen_2(A,B)) ) )).

fof(fc2_topgen_2,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(A)
     => ( ~ v3_struct_0(k5_topgen_2(A,B))
        & v1_pre_topc(k5_topgen_2(A,B))
        & v2_pre_topc(k5_topgen_2(A,B)) ) ) )).

fof(t2_topgen_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_pboole(B,u1_struct_0(A))
         => ( ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => m1_yellow_8(k1_funct_1(B,C),A,C) )
           => m1_cantor_1(k3_card_3(B),A) ) ) ) )).

fof(d1_topgen_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( v1_card_1(C)
             => ( C = k1_topgen_2(A,B)
              <=> ( ? [D] :
                      ( m1_yellow_8(D,A,B)
                      & C = k1_card_1(D) )
                  & ! [D] :
                      ( m1_yellow_8(D,A,B)
                     => r1_tarski(C,k1_card_1(D)) ) ) ) ) ) ) )).

fof(t3_topgen_2,axiom,(
    ! [A] :
      ( ! [B] :
          ( r2_hidden(B,A)
         => v1_card_1(B) )
     => v1_card_1(k3_tarski(A)) ) )).

fof(d2_topgen_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( v1_card_1(B)
         => ( B = k2_topgen_2(A)
          <=> ( ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => r1_tarski(k1_topgen_2(A,C),B) )
              & ! [C] :
                  ( v1_card_1(C)
                 => ( ! [D] :
                        ( m1_subset_1(D,u1_struct_0(A))
                       => r1_tarski(k1_topgen_2(A,D),C) )
                   => r1_tarski(B,C) ) ) ) ) ) ) )).

fof(t5_topgen_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => r1_tarski(k1_topgen_2(A,B),k2_topgen_2(A)) ) ) )).

fof(t6_topgen_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v1_frechet(A)
      <=> r1_tarski(k2_topgen_2(A),k3_card_1(np__0)) ) ) )).

fof(d3_topgen_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_pboole(B,u1_struct_0(A))
         => ( m1_topgen_2(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => m1_yellow_8(k1_funct_1(B,C),A,C) ) ) ) ) )).

fof(t7_topgen_2,axiom,(
    $true )).

fof(t8_topgen_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_yellow_8(D,A,B)
                 => ! [E] :
                      ( m1_yellow_8(E,A,C)
                     => ! [F] :
                          ~ ( r2_hidden(B,F)
                            & r2_hidden(F,E)
                            & ! [G] :
                                ( ( v3_pre_topc(G,A)
                                  & m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A))) )
                               => ~ ( r2_hidden(G,D)
                                    & r1_tarski(G,F) ) ) ) ) ) ) ) ) )).

fof(t9_topgen_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_yellow_8(C,A,B)
             => ! [D,E] :
                  ~ ( r2_hidden(D,C)
                    & r2_hidden(E,C)
                    & ! [F] :
                        ( ( v3_pre_topc(F,A)
                          & m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A))) )
                       => ~ ( r2_hidden(F,C)
                            & r1_tarski(F,k3_xboole_0(D,E)) ) ) ) ) ) ) )).

fof(t10_topgen_2,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,u1_struct_0(A))
             => ( r2_hidden(C,k6_pre_topc(A,B))
              <=> ! [D] :
                    ( m1_yellow_8(D,A,C)
                   => ! [E] :
                        ~ ( r2_hidden(E,D)
                          & r1_xboole_0(E,B) ) ) ) ) ) ) )).

fof(t11_topgen_2,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,u1_struct_0(A))
             => ( r2_hidden(C,k6_pre_topc(A,B))
              <=> ? [D] :
                    ( m1_yellow_8(D,A,C)
                    & ! [E] :
                        ~ ( r2_hidden(E,D)
                          & r1_xboole_0(E,B) ) ) ) ) ) ) )).

fof(t12_topgen_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v1_tops_2(B,A)
            & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) )
         => ? [C] :
              ( v1_tops_2(C,A)
              & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
              & r1_tarski(C,B)
              & k5_setfam_1(u1_struct_0(A),C) = k5_setfam_1(u1_struct_0(A),B)
              & r1_tarski(k1_card_1(C),k2_waybel23(A)) ) ) ) )).

fof(d4_topgen_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ( v1_topgen_2(A)
      <=> v1_finset_1(k2_waybel23(A)) ) ) )).

fof(t13_topgen_2,axiom,(
    ! [A] : k1_card_1(k3_pua2mss1(A)) = k1_card_1(A) )).

fof(t14_topgen_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_tdlat_3(A)
        & l1_pre_topc(A) )
     => ( m1_cantor_1(k3_pua2mss1(u1_struct_0(A)),A)
        & ! [B] :
            ( m1_cantor_1(B,A)
           => r1_tarski(k3_pua2mss1(u1_struct_0(A)),B) ) ) ) )).

fof(t15_topgen_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_tdlat_3(A)
        & l1_pre_topc(A) )
     => k2_waybel23(A) = k1_card_1(u1_struct_0(A)) ) )).

fof(t16_topgen_2,axiom,(
    $true )).

fof(t17_topgen_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v1_topgen_2(A)
        & l1_pre_topc(A) )
     => ? [B] :
          ( m1_cantor_1(B,A)
          & ? [C] :
              ( v1_funct_1(C)
              & v1_funct_2(C,u1_struct_0(A),u1_pre_topc(A))
              & m2_relset_1(C,u1_struct_0(A),u1_pre_topc(A))
              & B = k5_relset_1(u1_struct_0(A),u1_pre_topc(A),C)
              & ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( r2_hidden(D,k8_funct_2(u1_struct_0(A),u1_pre_topc(A),C,D))
                    & ! [E] :
                        ( ( v3_pre_topc(E,A)
                          & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) )
                       => ( r2_hidden(D,E)
                         => r1_tarski(k8_funct_2(u1_struct_0(A),u1_pre_topc(A),C,D),E) ) ) ) ) ) ) ) )).

fof(t18_topgen_2,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_cantor_1(B,A)
         => ! [C] :
              ( m1_cantor_1(C,A)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(A),u1_pre_topc(A))
                    & m2_relset_1(D,u1_struct_0(A),u1_pre_topc(A)) )
                 => ( ( B = k5_relset_1(u1_struct_0(A),u1_pre_topc(A),D)
                      & ! [E] :
                          ( m1_subset_1(E,u1_struct_0(A))
                         => ( r2_hidden(E,k1_funct_1(D,E))
                            & ! [F] :
                                ( ( v3_pre_topc(F,A)
                                  & m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A))) )
                               => ( r2_hidden(E,F)
                                 => r1_tarski(k1_funct_1(D,E),F) ) ) ) ) )
                   => r1_tarski(B,C) ) ) ) ) ) )).

fof(t19_topgen_2,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_cantor_1(B,A)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_pre_topc(A))
                & m2_relset_1(C,u1_struct_0(A),u1_pre_topc(A)) )
             => ( ( B = k5_relset_1(u1_struct_0(A),u1_pre_topc(A),C)
                  & ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ( r2_hidden(D,k1_funct_1(C,D))
                        & ! [E] :
                            ( ( v3_pre_topc(E,A)
                              & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) )
                           => ( r2_hidden(D,E)
                             => r1_tarski(k1_funct_1(C,D),E) ) ) ) ) )
               => k2_waybel23(A) = k1_card_1(B) ) ) ) ) )).

fof(t20_topgen_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_cantor_1(B,A)
         => ? [C] :
              ( m1_cantor_1(C,A)
              & r1_tarski(C,B)
              & k1_card_1(C) = k2_waybel23(A) ) ) ) )).

fof(t21_topgen_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_topgen_2(A,B))))
     => ( v3_pre_topc(C,k5_topgen_2(A,B))
      <=> ~ ( r2_hidden(B,C)
            & ~ v1_finset_1(k3_subset_1(u1_struct_0(k5_topgen_2(A,B)),C)) ) ) ) )).

fof(t22_topgen_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_topgen_2(A,B))))
     => ( v4_pre_topc(C,k5_topgen_2(A,B))
      <=> ~ ( r2_hidden(B,A)
            & ~ r2_hidden(B,C)
            & ~ v1_finset_1(C) ) ) ) )).

fof(t23_topgen_2,axiom,(
    ! [A,B,C] :
      ( r2_hidden(C,A)
     => ( v4_pre_topc(k1_tarski(C),k5_topgen_2(A,B))
        & m1_subset_1(k1_tarski(C),k1_zfmisc_1(u1_struct_0(k5_topgen_2(A,B)))) ) ) )).

fof(t24_topgen_2,axiom,(
    ! [A,B,C] :
      ( r2_hidden(C,A)
     => ( C = B
        | ( v3_pre_topc(k1_tarski(C),k5_topgen_2(A,B))
          & m1_subset_1(k1_tarski(C),k1_zfmisc_1(u1_struct_0(k5_topgen_2(A,B)))) ) ) ) )).

fof(t25_topgen_2,axiom,(
    ! [A,B] :
      ( ~ v1_finset_1(A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_topgen_2(A,B))))
         => ~ ( C = k1_tarski(B)
              & v3_pre_topc(C,k5_topgen_2(A,B)) ) ) ) )).

fof(t26_topgen_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_topgen_2(A,B))))
     => ( v1_finset_1(C)
       => k6_pre_topc(k5_topgen_2(A,B),C) = C ) ) )).

fof(t27_topgen_2,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)))
         => ( ~ v4_pre_topc(B,A)
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ( v4_pre_topc(k4_subset_1(u1_struct_0(A),B,k1_struct_0(A,C)),A)
                 => k6_pre_topc(A,B) = k4_subset_1(u1_struct_0(A),B,k1_struct_0(A,C)) ) ) ) ) ) )).

fof(t28_topgen_2,axiom,(
    ! [A,B] :
      ( r2_hidden(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_topgen_2(A,B))))
         => ( ~ v1_finset_1(C)
           => k6_pre_topc(k5_topgen_2(A,B),C) = k2_xboole_0(C,k1_tarski(B)) ) ) ) )).

fof(t29_topgen_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_topgen_2(A,B))))
     => ( v1_finset_1(k3_subset_1(u1_struct_0(k5_topgen_2(A,B)),C))
       => k1_tops_1(k5_topgen_2(A,B),C) = C ) ) )).

fof(t30_topgen_2,axiom,(
    ! [A,B] :
      ( r2_hidden(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_topgen_2(A,B))))
         => ( ~ v1_finset_1(k3_subset_1(u1_struct_0(k5_topgen_2(A,B)),C))
           => k1_tops_1(k5_topgen_2(A,B),C) = k4_xboole_0(C,k1_tarski(B)) ) ) ) )).

fof(t32_topgen_2,axiom,(
    ! [A] :
      ( ~ v1_finset_1(A)
     => k1_card_1(k5_finsub_1(A)) = k1_card_1(A) ) )).

fof(t35_topgen_2,axiom,(
    ! [A] :
      ( ~ v1_finset_1(A)
     => ! [B,C] :
          ( m1_cantor_1(C,k5_topgen_2(A,B))
         => r1_tarski(k1_card_1(A),k1_card_1(C)) ) ) )).

fof(t36_topgen_2,axiom,(
    ! [A] :
      ( ~ v1_finset_1(A)
     => ! [B] : k2_waybel23(k5_topgen_2(A,B)) = k1_card_1(A) ) )).

fof(t40_topgen_2,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(k1_pcomps_1(A))))
     => ~ ( ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
             => ( r2_hidden(C,B)
               => ( v2_pre_topc(g1_pre_topc(A,C))
                  & l1_pre_topc(g1_pre_topc(A,C)) ) ) )
          & ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
             => ~ ( C = k8_setfam_1(k1_pcomps_1(A),B)
                  & v2_pre_topc(g1_pre_topc(A,C))
                  & l1_pre_topc(g1_pre_topc(A,C))
                  & ! [D] :
                      ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(A)))
                     => ( r2_hidden(D,B)
                       => m2_yellow_9(g1_pre_topc(A,D),g1_pre_topc(A,C)) ) )
                  & ! [D] :
                      ( ( v2_pre_topc(D)
                        & l1_pre_topc(D) )
                     => ( ( u1_struct_0(D) = A
                          & ! [E] :
                              ( m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(A)))
                             => ( r2_hidden(E,B)
                               => m2_yellow_9(g1_pre_topc(A,E),D) ) ) )
                       => m2_yellow_9(g1_pre_topc(A,C),D) ) ) ) ) ) ) )).

fof(t41_topgen_2,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(k1_pcomps_1(A))))
     => ? [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
          & C = k1_cantor_1(A,k2_cantor_1(A,k5_setfam_1(k1_pcomps_1(A),B)))
          & v2_pre_topc(g1_pre_topc(A,C))
          & l1_pre_topc(g1_pre_topc(A,C))
          & ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(A)))
             => ( r2_hidden(D,B)
               => m2_yellow_9(g1_pre_topc(A,C),g1_pre_topc(A,D)) ) )
          & ! [D] :
              ( ( v2_pre_topc(D)
                & l1_pre_topc(D) )
             => ( ( u1_struct_0(D) = A
                  & ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(A)))
                     => ( r2_hidden(E,B)
                       => m2_yellow_9(D,g1_pre_topc(A,E)) ) ) )
               => m2_yellow_9(D,g1_pre_topc(A,C)) ) ) ) ) )).

fof(dt_m1_topgen_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_topgen_2(B,A)
         => m1_pboole(B,u1_struct_0(A)) ) ) )).

fof(existence_m1_topgen_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ? [B] : m1_topgen_2(B,A) ) )).

fof(dt_k1_topgen_2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A)
        & m1_subset_1(B,u1_struct_0(A)) )
     => v1_card_1(k1_topgen_2(A,B)) ) )).

fof(dt_k2_topgen_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A) )
     => v1_card_1(k2_topgen_2(A)) ) )).

fof(dt_k3_topgen_2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & m1_topgen_2(B,A) )
     => m1_cantor_1(k3_topgen_2(A,B),A) ) )).

fof(redefinition_k3_topgen_2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & m1_topgen_2(B,A) )
     => k3_topgen_2(A,B) = k3_card_3(B) ) )).

fof(dt_k4_topgen_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & m1_topgen_2(B,A)
        & m1_subset_1(C,u1_struct_0(A)) )
     => m1_yellow_8(k4_topgen_2(A,B,C),A,C) ) )).

fof(redefinition_k4_topgen_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & m1_topgen_2(B,A)
        & m1_subset_1(C,u1_struct_0(A)) )
     => k4_topgen_2(A,B,C) = k1_funct_1(B,C) ) )).

fof(dt_k5_topgen_2,axiom,(
    ! [A,B] :
      ( v1_pre_topc(k5_topgen_2(A,B))
      & l1_pre_topc(k5_topgen_2(A,B)) ) )).

fof(t1_topgen_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_cantor_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => m1_yellow_8(a_3_0_topgen_2(A,B,C),A,C) ) ) ) )).

fof(t4_topgen_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A) )
     => k2_topgen_2(A) = k3_tarski(a_1_0_topgen_2(A)) ) )).

fof(d5_topgen_2,axiom,(
    ! [A,B,C] :
      ( ( v1_pre_topc(C)
        & l1_pre_topc(C) )
     => ( C = k5_topgen_2(A,B)
      <=> ( u1_struct_0(C) = A
          & u1_pre_topc(C) = k2_xboole_0(a_2_0_topgen_2(A,B),a_1_1_topgen_2(A)) ) ) ) )).

fof(t31_topgen_2,axiom,(
    ! [A,B] :
    ? [C] :
      ( m1_cantor_1(C,k5_topgen_2(A,B))
      & C = k2_xboole_0(k4_xboole_0(k3_pua2mss1(A),k1_tarski(k1_tarski(B))),a_1_1_topgen_2(A)) ) )).

fof(t33_topgen_2,axiom,(
    ! [A] :
      ( ~ v1_finset_1(A)
     => k1_card_1(a_1_2_topgen_2(A)) = k1_card_1(A) ) )).

fof(t34_topgen_2,axiom,(
    ! [A] :
      ( ~ v1_finset_1(A)
     => ! [B,C] :
          ( m1_cantor_1(C,k5_topgen_2(A,B))
         => ( C = k2_xboole_0(k4_xboole_0(k3_pua2mss1(A),k1_tarski(k1_tarski(B))),a_1_2_topgen_2(A))
           => k1_card_1(C) = k1_card_1(A) ) ) ) )).

fof(t37_topgen_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
        ? [C] :
          ( m2_cantor_1(C,k5_topgen_2(A,B))
          & C = k2_xboole_0(k4_xboole_0(k3_pua2mss1(A),k1_tarski(k1_tarski(B))),a_1_3_topgen_2(A)) ) ) )).

fof(t38_topgen_2,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] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(B))) )
             => ( ! [D] :
                    ( r2_hidden(D,C)
                   => v1_finset_1(k4_xboole_0(B,D)) )
               => k6_pre_topc(A,k5_setfam_1(u1_struct_0(A),B)) = k2_xboole_0(k5_setfam_1(u1_struct_0(A),k3_pcomps_1(A,B)),k1_setfam_1(a_3_1_topgen_2(A,B,C))) ) ) ) ) )).

fof(t39_topgen_2,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))))
         => k6_pre_topc(A,k5_setfam_1(u1_struct_0(A),B)) = k2_xboole_0(k5_setfam_1(u1_struct_0(A),k3_pcomps_1(A,B)),k1_setfam_1(a_2_1_topgen_2(A,B))) ) ) )).

fof(fraenkel_a_3_0_topgen_2,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & v2_pre_topc(B)
        & l1_pre_topc(B)
        & m1_cantor_1(C,B)
        & m1_subset_1(D,u1_struct_0(B)) )
     => ( r2_hidden(A,a_3_0_topgen_2(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
            & A = E
            & r2_hidden(D,E)
            & r2_hidden(E,C) ) ) ) )).

fof(fraenkel_a_1_0_topgen_2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l1_pre_topc(B) )
     => ( r2_hidden(A,a_1_0_topgen_2(B))
      <=> ? [C] :
            ( m1_subset_1(C,u1_struct_0(B))
            & A = k1_topgen_2(B,C) ) ) ) )).

fof(fraenkel_a_2_0_topgen_2,axiom,(
    ! [A,B,C] :
      ( r2_hidden(A,a_2_0_topgen_2(B,C))
    <=> ? [D] :
          ( m1_subset_1(D,k1_zfmisc_1(B))
          & A = D
          & ~ r2_hidden(C,D) ) ) )).

fof(fraenkel_a_1_1_topgen_2,axiom,(
    ! [A,B] :
      ( r2_hidden(A,a_1_1_topgen_2(B))
    <=> ? [C] :
          ( m1_subset_1(C,k1_zfmisc_1(B))
          & A = k3_subset_1(B,C)
          & v1_finset_1(C) ) ) )).

fof(fraenkel_a_1_2_topgen_2,axiom,(
    ! [A,B] :
      ( ~ v1_finset_1(B)
     => ( r2_hidden(A,a_1_2_topgen_2(B))
      <=> ? [C] :
            ( m1_subset_1(C,k1_zfmisc_1(B))
            & A = k3_subset_1(B,C)
            & v1_finset_1(C) ) ) ) )).

fof(fraenkel_a_1_3_topgen_2,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ( r2_hidden(A,a_1_3_topgen_2(B))
      <=> ? [C] :
            ( m1_subset_1(C,B)
            & A = k3_subset_1(B,k6_domain_1(B,C)) ) ) ) )).

fof(fraenkel_a_3_1_topgen_2,axiom,(
    ! [A,B,C,D] :
      ( ( v2_pre_topc(B)
        & l1_pre_topc(B)
        & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B))))
        & ~ v1_xboole_0(D)
        & m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(C))) )
     => ( r2_hidden(A,a_3_1_topgen_2(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B))))
            & A = k6_pre_topc(B,k5_setfam_1(u1_struct_0(B),E))
            & r2_hidden(E,D) ) ) ) )).

fof(fraenkel_a_2_1_topgen_2,axiom,(
    ! [A,B,C] :
      ( ( v2_pre_topc(B)
        & l1_pre_topc(B)
        & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B)))) )
     => ( r2_hidden(A,a_2_1_topgen_2(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B))))
            & A = k6_pre_topc(B,k5_setfam_1(u1_struct_0(B),D))
            & r1_tarski(D,C)
            & v1_finset_1(k6_subset_1(k1_zfmisc_1(u1_struct_0(B)),C,D)) ) ) ) )).
%------------------------------------------------------------------------------