SET007 Axioms: SET007+217.ax


%------------------------------------------------------------------------------
% File     : SET007+217 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Compact 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    : compts_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   33 (   8 unit)
%            Number of atoms       :  232 (  21 equality)
%            Maximal formula depth :   23 (   9 average)
%            Number of connectives :  229 (  30 ~  ;   2  |;  94  &)
%                                         (  13 <=>;  90 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   30 (   1 propositional; 0-3 arity)
%            Number of functors    :   15 (   1 constant; 0-4 arity)
%            Number of variables   :   74 (   0 singleton;  74 !;   0 ?)
%            Maximal term depth    :    4 (   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(d1_compts_1,axiom,(
    ! [A] :
      ( l1_struct_0(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)))
             => ( r1_compts_1(A,B,C)
              <=> r1_tarski(C,k5_setfam_1(u1_struct_0(A),B)) ) ) ) ) )).

fof(d2_compts_1,axiom,(
    ! [A] :
      ( v1_compts_1(A)
    <=> ( A != k1_xboole_0
        & ! [B] :
            ~ ( B != k1_xboole_0
              & r1_tarski(B,A)
              & v1_finset_1(B)
              & k1_setfam_1(B) = k1_xboole_0 ) ) ) )).

fof(d3_compts_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ( v2_compts_1(A)
      <=> ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
           => ~ ( r1_pre_topc(A,B)
                & v1_tops_2(B,A)
                & ! [C] :
                    ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
                   => ~ ( r1_tarski(C,B)
                        & r1_pre_topc(A,C)
                        & v1_finset_1(C) ) ) ) ) ) ) )).

fof(d4_compts_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ( v3_compts_1(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ~ ( B != C
                    & ! [D] :
                        ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                       => ! [E] :
                            ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                           => ~ ( v3_pre_topc(D,A)
                                & v3_pre_topc(E,A)
                                & r2_hidden(B,D)
                                & r2_hidden(C,E)
                                & r1_xboole_0(D,E) ) ) ) ) ) ) ) ) )).

fof(d5_compts_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ( v4_compts_1(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
               => ~ ( C != k1_xboole_0
                    & v4_pre_topc(C,A)
                    & r2_hidden(B,k3_subset_1(u1_struct_0(A),C))
                    & ! [D] :
                        ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                       => ! [E] :
                            ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                           => ~ ( v3_pre_topc(D,A)
                                & v3_pre_topc(E,A)
                                & r2_hidden(B,D)
                                & r1_tarski(C,E)
                                & r1_xboole_0(D,E) ) ) ) ) ) ) ) ) )).

fof(d6_compts_1,axiom,(
    ! [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 != k1_xboole_0
                    & v4_pre_topc(B,A)
                    & v4_pre_topc(C,A)
                    & r1_xboole_0(B,C)
                    & ! [D] :
                        ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                       => ! [E] :
                            ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(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(d7_compts_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v6_compts_1(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
               => ~ ( r1_compts_1(A,C,B)
                    & v1_tops_2(C,A)
                    & ! [D] :
                        ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
                       => ~ ( r1_tarski(D,C)
                            & r1_compts_1(A,D,B)
                            & v1_finset_1(D) ) ) ) ) ) ) ) )).

fof(t1_compts_1,axiom,(
    $true )).

fof(t2_compts_1,axiom,(
    $true )).

fof(t3_compts_1,axiom,(
    $true )).

fof(t4_compts_1,axiom,(
    $true )).

fof(t5_compts_1,axiom,(
    $true )).

fof(t6_compts_1,axiom,(
    $true )).

fof(t7_compts_1,axiom,(
    $true )).

fof(t8_compts_1,axiom,(
    $true )).

fof(t9_compts_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => v6_compts_1(k1_pre_topc(A),A) ) )).

fof(t10_compts_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ( v2_compts_1(A)
      <=> v6_compts_1(k2_pre_topc(A),A) ) ) )).

fof(t11_compts_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_pre_topc(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( r1_tarski(C,k2_pre_topc(B))
               => ( v6_compts_1(C,A)
                <=> ! [D] :
                      ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
                     => ( D = C
                       => v6_compts_1(D,B) ) ) ) ) ) ) ) )).

fof(t12_compts_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( ( B = k1_xboole_0
             => ( v6_compts_1(B,A)
              <=> v2_compts_1(k3_pre_topc(A,B)) ) )
            & ( v2_pre_topc(A)
             => ( B = k1_xboole_0
                | ( v6_compts_1(B,A)
                <=> v2_compts_1(k3_pre_topc(A,B)) ) ) ) ) ) ) )).

fof(t13_compts_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v2_compts_1(A)
      <=> ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
           => ~ ( v1_compts_1(B)
                & v2_tops_2(B,A)
                & k6_setfam_1(u1_struct_0(A),B) = k1_xboole_0 ) ) ) ) )).

fof(t14_compts_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v2_compts_1(A)
      <=> ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
           => ~ ( B != k1_xboole_0
                & v2_tops_2(B,A)
                & k6_setfam_1(u1_struct_0(A),B) = k1_xboole_0
                & ! [C] :
                    ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
                   => ~ ( C != k1_xboole_0
                        & r1_tarski(C,B)
                        & v1_finset_1(C)
                        & k6_setfam_1(u1_struct_0(A),C) = k1_xboole_0 ) ) ) ) ) ) )).

fof(t15_compts_1,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v3_compts_1(A)
       => ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
           => ( v6_compts_1(B,A)
             => ( B = k1_xboole_0
                | ! [C] :
                    ( m1_subset_1(C,u1_struct_0(A))
                   => ~ ( r2_hidden(C,k3_subset_1(u1_struct_0(A),B))
                        & ! [D] :
                            ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                           => ! [E] :
                                ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                               => ~ ( v3_pre_topc(D,A)
                                    & v3_pre_topc(E,A)
                                    & r2_hidden(C,D)
                                    & r1_tarski(B,E)
                                    & r1_xboole_0(D,E) ) ) ) ) ) ) ) ) ) ) )).

fof(t16_compts_1,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( ( v3_compts_1(A)
              & v6_compts_1(B,A) )
           => v4_pre_topc(B,A) ) ) ) )).

fof(t17_compts_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( ( v2_compts_1(A)
              & v4_pre_topc(B,A) )
           => v6_compts_1(B,A) ) ) ) )).

fof(t18_compts_1,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)))
             => ( ( v6_compts_1(B,A)
                  & r1_tarski(C,B)
                  & v4_pre_topc(C,A) )
               => v6_compts_1(C,A) ) ) ) ) )).

fof(t19_compts_1,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)))
             => ( ( v6_compts_1(B,A)
                  & v6_compts_1(C,A) )
               => v6_compts_1(k4_subset_1(u1_struct_0(A),B,C),A) ) ) ) ) )).

fof(t20_compts_1,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)))
             => ( ( v3_compts_1(A)
                  & v6_compts_1(B,A)
                  & v6_compts_1(C,A) )
               => v6_compts_1(k5_subset_1(u1_struct_0(A),B,C),A) ) ) ) ) )).

fof(t21_compts_1,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( ( v3_compts_1(A)
          & v2_compts_1(A) )
       => v4_compts_1(A) ) ) )).

fof(t22_compts_1,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( ( v3_compts_1(A)
          & v2_compts_1(A) )
       => v5_compts_1(A) ) ) )).

fof(t23_compts_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( ( ~ v3_struct_0(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)) )
             => ( ( v2_compts_1(A)
                  & v5_pre_topc(C,A,B)
                  & k2_relat_1(C) = k2_pre_topc(B) )
               => v2_compts_1(B) ) ) ) ) )).

fof(t24_compts_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & l1_pre_topc(C) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(A),u1_struct_0(C))
                    & m2_relset_1(D,u1_struct_0(A),u1_struct_0(C)) )
                 => ( ( v5_pre_topc(D,A,C)
                      & k2_relat_1(D) = k2_pre_topc(C)
                      & v6_compts_1(B,A) )
                   => v6_compts_1(k4_pre_topc(A,C,D,B),C) ) ) ) ) ) )).

fof(t25_compts_1,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(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)) )
             => ( ( v2_compts_1(A)
                  & v3_compts_1(B)
                  & k2_relat_1(C) = k2_pre_topc(B)
                  & v5_pre_topc(C,A,B) )
               => ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                   => ( v4_pre_topc(D,A)
                     => v4_pre_topc(k4_pre_topc(A,B,C,D),B) ) ) ) ) ) ) )).

fof(t26_compts_1,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(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)) )
             => ( ( v2_compts_1(A)
                  & v3_compts_1(B)
                  & k1_relat_1(C) = k2_pre_topc(A)
                  & k2_relat_1(C) = k2_pre_topc(B)
                  & v2_funct_1(C)
                  & v5_pre_topc(C,A,B) )
               => v3_tops_2(C,A,B) ) ) ) ) )).
%------------------------------------------------------------------------------