SET007 Axioms: SET007+256.ax


%------------------------------------------------------------------------------
% File     : SET007+256 : TPTP v9.0.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Locally Connected 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    : connsp_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   45 (   7 unt;   0 def)
%            Number of atoms       :  355 (  17 equ)
%            Maximal formula atoms :   23 (   7 avg)
%            Number of connectives :  378 (  68   ~;   0   |; 154   &)
%                                         (  17 <=>; 139  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   27 (   9 avg)
%            Maximal term depth    :    4 (   1 avg)
%            Number of predicates  :   23 (  21 usr;   1 prp; 0-3 aty)
%            Number of functors    :   13 (  13 usr;   1 con; 0-3 aty)
%            Number of variables   :  130 ( 126   !;   4   ?)
% 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(d1_connsp_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,k1_zfmisc_1(u1_struct_0(A)))
             => ( m1_connsp_2(C,A,B)
              <=> r2_hidden(B,k1_tops_1(A,C)) ) ) ) ) ).

fof(d2_connsp_2,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)))
             => ( m2_connsp_2(C,A,B)
              <=> r1_tarski(B,k1_tops_1(A,C)) ) ) ) ) ).

fof(t1_connsp_2,axiom,
    $true ).

fof(t2_connsp_2,axiom,
    $true ).

fof(t3_connsp_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,k1_zfmisc_1(u1_struct_0(A)))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => ( ( m1_connsp_2(C,A,B)
                      & m1_connsp_2(D,A,B) )
                   => m1_connsp_2(k4_subset_1(u1_struct_0(A),C,D),A,B) ) ) ) ) ) ).

fof(t4_connsp_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,k1_zfmisc_1(u1_struct_0(A)))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => ( ( m1_connsp_2(C,A,B)
                      & m1_connsp_2(D,A,B) )
                  <=> m1_connsp_2(k5_subset_1(u1_struct_0(A),C,D),A,B) ) ) ) ) ) ).

fof(t5_connsp_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))
             => ( ( v3_pre_topc(B,A)
                  & r2_hidden(C,B) )
               => m1_connsp_2(B,A,C) ) ) ) ) ).

fof(t6_connsp_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))
             => ( m1_connsp_2(B,A,C)
               => r2_hidden(C,B) ) ) ) ) ).

fof(t7_connsp_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,k1_zfmisc_1(u1_struct_0(A)))
             => ~ ( m1_connsp_2(C,A,B)
                  & ! [D] :
                      ( ( ~ v1_xboole_0(D)
                        & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
                     => ~ ( m1_connsp_2(D,A,B)
                          & v3_pre_topc(D,A)
                          & r1_tarski(D,C) ) ) ) ) ) ) ).

fof(t8_connsp_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,k1_zfmisc_1(u1_struct_0(A)))
             => ( m1_connsp_2(C,A,B)
              <=> ? [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                    & v3_pre_topc(D,A)
                    & r1_tarski(D,C)
                    & r2_hidden(B,D) ) ) ) ) ) ).

fof(t9_connsp_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)))
         => ( v3_pre_topc(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ~ ( r2_hidden(C,B)
                    & ! [D] :
                        ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                       => ~ ( m1_connsp_2(D,A,C)
                            & r1_tarski(D,B) ) ) ) ) ) ) ) ).

fof(t10_connsp_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,k1_zfmisc_1(u1_struct_0(A)))
             => ( m2_connsp_2(C,A,k1_struct_0(A,B))
              <=> m1_connsp_2(C,A,B) ) ) ) ) ).

fof(t11_connsp_2,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))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k3_pre_topc(A,B)))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k3_pre_topc(A,B))))
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(A))
                         => ( ( v3_pre_topc(B,A)
                              & m1_connsp_2(D,k3_pre_topc(A,B),C)
                              & D = E
                              & C = F )
                           => m1_connsp_2(E,A,F) ) ) ) ) ) ) ) ).

fof(t12_connsp_2,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))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k3_pre_topc(A,B)))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k3_pre_topc(A,B))))
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(A))
                         => ( ( m1_connsp_2(E,A,F)
                              & D = E
                              & C = F )
                           => m1_connsp_2(D,k3_pre_topc(A,B),C) ) ) ) ) ) ) ) ).

fof(t13_connsp_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,k1_zfmisc_1(u1_struct_0(A)))
             => ( ( r3_connsp_1(A,B)
                  & r1_tarski(B,C) )
               => r4_connsp_1(A,C,B) ) ) ) ) ).

fof(t14_connsp_2,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & m1_pre_topc(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ( C = D
                   => r1_tarski(k1_connsp_1(B,D),k1_connsp_1(A,C)) ) ) ) ) ) ).

fof(d3_connsp_2,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( r1_connsp_2(A,B)
          <=> ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
               => ~ ( m1_connsp_2(C,A,B)
                    & ! [D] :
                        ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                       => ~ ( m1_connsp_2(D,A,B)
                            & v2_connsp_1(D,A)
                            & r1_tarski(D,C) ) ) ) ) ) ) ) ).

fof(d4_connsp_2,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v1_connsp_2(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => r1_connsp_2(A,B) ) ) ) ).

fof(d5_connsp_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_connsp_2(A,B,C)
              <=> ! [D] :
                    ( ( ~ v1_xboole_0(D)
                      & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
                   => ~ ( B = D
                        & ! [E] :
                            ( m1_subset_1(E,u1_struct_0(k3_pre_topc(A,D)))
                           => ~ ( E = C
                                & r1_connsp_2(k3_pre_topc(A,D),E) ) ) ) ) ) ) ) ) ).

fof(d6_connsp_2,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_connsp_2(B,A)
          <=> v1_connsp_2(k3_pre_topc(A,B)) ) ) ) ).

fof(t15_connsp_2,axiom,
    $true ).

fof(t16_connsp_2,axiom,
    $true ).

fof(t17_connsp_2,axiom,
    $true ).

fof(t18_connsp_2,axiom,
    $true ).

fof(t19_connsp_2,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( r1_connsp_2(A,B)
          <=> ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
               => ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                   => ( ( m1_connsp_2(C,A,B)
                        & r4_connsp_1(A,C,D)
                        & r2_hidden(B,D) )
                     => m1_connsp_2(D,A,B) ) ) ) ) ) ) ).

fof(t20_connsp_2,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( r1_connsp_2(A,B)
          <=> ! [C] :
                ( ( ~ v1_xboole_0(C)
                  & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
               => ~ ( v3_pre_topc(C,A)
                    & r2_hidden(B,C)
                    & ! [D] :
                        ( m1_subset_1(D,u1_struct_0(k3_pre_topc(A,C)))
                       => ~ ( D = B
                            & r2_hidden(B,k1_tops_1(k3_pre_topc(A,C),k1_connsp_1(k3_pre_topc(A,C),D))) ) ) ) ) ) ) ) ).

fof(t21_connsp_2,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v1_connsp_2(A)
       => ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
           => ( r3_connsp_1(A,B)
             => v3_pre_topc(B,A) ) ) ) ) ).

fof(t22_connsp_2,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( r1_connsp_2(A,B)
           => ! [C] :
                ( ( ~ v1_xboole_0(C)
                  & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
               => ( ( v3_pre_topc(C,A)
                    & r2_hidden(B,C) )
                 => r2_connsp_2(A,C,B) ) ) ) ) ) ).

fof(t23_connsp_2,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v1_connsp_2(A)
       => ! [B] :
            ( ( ~ v1_xboole_0(B)
              & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
           => ( v3_pre_topc(B,A)
             => v2_connsp_2(B,A) ) ) ) ) ).

fof(t24_connsp_2,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v1_connsp_2(A)
      <=> ! [B] :
            ( ( ~ v1_xboole_0(B)
              & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
           => ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
               => ( ( v3_pre_topc(B,A)
                    & r4_connsp_1(A,B,C) )
                 => v3_pre_topc(C,A) ) ) ) ) ) ).

fof(t25_connsp_2,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v1_connsp_2(A)
       => ! [B] :
            ( ( ~ v1_xboole_0(B)
              & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
           => ! [C] :
                ( ( ~ v1_xboole_0(C)
                  & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k3_pre_topc(A,B)))) )
               => ~ ( v2_connsp_1(C,k3_pre_topc(A,B))
                    & v3_pre_topc(C,k3_pre_topc(A,B))
                    & ! [D] :
                        ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                       => ~ ( v3_pre_topc(D,A)
                            & v2_connsp_1(D,A)
                            & C = k5_subset_1(u1_struct_0(A),B,D) ) ) ) ) ) ) ) ).

fof(t26_connsp_2,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v5_compts_1(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)))
               => ~ ( B != k1_xboole_0
                    & C != k2_pre_topc(A)
                    & r1_tarski(B,C)
                    & v4_pre_topc(B,A)
                    & v3_pre_topc(C,A)
                    & ! [D] :
                        ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                       => ~ ( v3_pre_topc(D,A)
                            & r1_tarski(B,D)
                            & r1_tarski(k6_pre_topc(A,D),C) ) ) ) ) ) ) ) ).

fof(t27_connsp_2,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( ( v1_connsp_2(A)
          & v5_compts_1(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)))
               => ~ ( B != k1_xboole_0
                    & C != k1_xboole_0
                    & v4_pre_topc(B,A)
                    & v4_pre_topc(C,A)
                    & r1_xboole_0(B,C)
                    & v2_connsp_1(B,A)
                    & v2_connsp_1(C,A)
                    & ! [D] :
                        ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                       => ! [E] :
                            ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                           => ~ ( v2_connsp_1(D,A)
                                & v2_connsp_1(E,A)
                                & v3_pre_topc(D,A)
                                & v3_pre_topc(E,A)
                                & r1_tarski(B,D)
                                & r1_tarski(C,E)
                                & r1_xboole_0(D,E) ) ) ) ) ) ) ) ) ).

fof(t28_connsp_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,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)
                      <=> ( v3_pre_topc(D,A)
                          & v4_pre_topc(D,A)
                          & r2_hidden(B,D) ) ) )
                  & C = k1_xboole_0 ) ) ) ) ).

fof(d7_connsp_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,k1_zfmisc_1(u1_struct_0(A)))
             => ( C = k1_connsp_2(A,B)
              <=> ? [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
                    & ! [E] :
                        ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                       => ( r2_hidden(E,D)
                        <=> ( v3_pre_topc(E,A)
                            & v4_pre_topc(E,A)
                            & r2_hidden(B,E) ) ) )
                    & k6_setfam_1(u1_struct_0(A),D) = C ) ) ) ) ) ).

fof(t29_connsp_2,axiom,
    $true ).

fof(t30_connsp_2,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => r2_hidden(B,k1_connsp_2(A,B)) ) ) ).

fof(t31_connsp_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,k1_zfmisc_1(u1_struct_0(A)))
             => ( ( v3_pre_topc(C,A)
                  & v4_pre_topc(C,A)
                  & r2_hidden(B,C)
                  & r1_tarski(C,k1_connsp_2(A,B)) )
               => C = k1_connsp_2(A,B) ) ) ) ) ).

fof(t32_connsp_2,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => v4_pre_topc(k1_connsp_2(A,B),A) ) ) ).

fof(t33_connsp_2,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => r1_tarski(k1_connsp_1(A,B),k1_connsp_2(A,B)) ) ) ).

fof(dt_m1_connsp_2,axiom,
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & m1_subset_1(B,u1_struct_0(A)) )
     => ! [C] :
          ( m1_connsp_2(C,A,B)
         => m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) ) ) ).

fof(existence_m1_connsp_2,axiom,
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & m1_subset_1(B,u1_struct_0(A)) )
     => ? [C] : m1_connsp_2(C,A,B) ) ).

fof(dt_m2_connsp_2,axiom,
    ! [A,B] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ! [C] :
          ( m2_connsp_2(C,A,B)
         => m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) ) ) ).

fof(existence_m2_connsp_2,axiom,
    ! [A,B] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ? [C] : m2_connsp_2(C,A,B) ) ).

fof(dt_k1_connsp_2,axiom,
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & m1_subset_1(B,u1_struct_0(A)) )
     => m1_subset_1(k1_connsp_2(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).

%------------------------------------------------------------------------------