SET007 Axioms: SET007+399.ax


%------------------------------------------------------------------------------
% File     : SET007+399 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Tzero 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    : t_0topsp [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   31 (   0 unit)
%            Number of atoms       :  237 (  19 equality)
%            Maximal formula depth :   18 (   8 average)
%            Number of connectives :  248 (  42 ~  ;   3  |; 110  &)
%                                         (  11 <=>;  82 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   23 (   0 propositional; 1-3 arity)
%            Number of functors    :   24 (   0 constant; 1-5 arity)
%            Number of variables   :   79 (   0 singleton;  73 !;   6 ?)
%            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(fc1_t_0topsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( ~ v3_struct_0(k3_t_0topsp(A))
        & v2_pre_topc(k3_t_0topsp(A)) ) ) )).

fof(rc1_t_0topsp,axiom,(
    ? [A] :
      ( l1_pre_topc(A)
      & ~ v3_struct_0(A)
      & v2_pre_topc(A)
      & v2_t_0topsp(A) ) )).

fof(t1_t_0topsp,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m2_relset_1(C,A,B)
             => ! [D] :
                  ( m2_relset_1(D,A,B)
                 => ( ! [E] :
                        ( m1_subset_1(E,A)
                       => ! [F] :
                            ( m1_subset_1(F,B)
                           => ( r2_hidden(k4_tarski(E,F),C)
                            <=> r2_hidden(k4_tarski(E,F),D) ) ) )
                   => C = D ) ) ) ) ) )).

fof(t2_t_0topsp,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,A,B)
                & m2_relset_1(C,A,B) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(A))
                 => ( ! [E] :
                        ( m1_subset_1(E,A)
                       => ! [F] :
                            ( m1_subset_1(F,A)
                           => ( ( r2_hidden(E,D)
                                & k8_funct_2(A,B,C,E) = k8_funct_2(A,B,C,F) )
                             => r2_hidden(F,D) ) ) )
                   => k3_funct_2(A,B,C,k2_funct_2(A,B,C,D)) = D ) ) ) ) ) )).

fof(d1_t_0topsp,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( l1_pre_topc(B)
         => ( r1_t_0topsp(A,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))
                & v3_tops_2(C,A,B) ) ) ) ) )).

fof(d2_t_0topsp,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)) )
             => ( v1_t_0topsp(C,A,B)
              <=> ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                   => ( v3_pre_topc(D,A)
                     => v3_pre_topc(k4_pre_topc(A,B,C,D),B) ) ) ) ) ) ) )).

fof(d3_t_0topsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v3_relat_2(B)
            & v8_relat_2(B)
            & v1_partfun1(B,u1_struct_0(A),u1_struct_0(A))
            & m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
         => ( B = k1_t_0topsp(A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ( r2_hidden(k4_tarski(C,D),B)
                    <=> ! [E] :
                          ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                         => ( v3_pre_topc(E,A)
                           => ( r2_hidden(C,E)
                            <=> r2_hidden(D,E) ) ) ) ) ) ) ) ) ) )).

fof(d4_t_0topsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A) )
     => k2_t_0topsp(A) = k8_eqrel_1(u1_struct_0(A),k1_t_0topsp(A)) ) )).

fof(d5_t_0topsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => k3_t_0topsp(A) = k16_borsuk_1(A,k2_t_0topsp(A)) ) )).

fof(d6_t_0topsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => k4_t_0topsp(A) = k17_borsuk_1(A,k2_t_0topsp(A)) ) )).

fof(t3_t_0topsp,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,k8_funct_2(u1_struct_0(A),u1_struct_0(k3_t_0topsp(A)),k4_t_0topsp(A),B)) ) ) )).

fof(t4_t_0topsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( k1_relat_1(k4_t_0topsp(A)) = u1_struct_0(A)
        & r1_tarski(k2_relat_1(k4_t_0topsp(A)),u1_struct_0(k3_t_0topsp(A))) ) ) )).

fof(t6_t_0topsp,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(k3_t_0topsp(A))))
         => ( v3_pre_topc(B,k3_t_0topsp(A))
          <=> r2_hidden(k3_tarski(B),u1_pre_topc(A)) ) ) ) )).

fof(t7_t_0topsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k3_t_0topsp(A)))
        <=> ? [C] :
              ( m1_subset_1(C,u1_struct_0(A))
              & B = k6_eqrel_1(u1_struct_0(A),k1_t_0topsp(A),C) ) ) ) )).

fof(t8_t_0topsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k8_funct_2(u1_struct_0(A),u1_struct_0(k3_t_0topsp(A)),k4_t_0topsp(A),B) = k6_eqrel_1(u1_struct_0(A),k1_t_0topsp(A),B) ) ) )).

fof(t9_t_0topsp,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))
             => ( k8_funct_2(u1_struct_0(A),u1_struct_0(k3_t_0topsp(A)),k4_t_0topsp(A),C) = k8_funct_2(u1_struct_0(A),u1_struct_0(k3_t_0topsp(A)),k4_t_0topsp(A),B)
              <=> r2_hidden(k8_borsuk_1(A,A,C,B),k1_t_0topsp(A)) ) ) ) ) )).

fof(t10_t_0topsp,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))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ( ( r2_hidden(C,B)
                        & k8_funct_2(u1_struct_0(A),u1_struct_0(k3_t_0topsp(A)),k4_t_0topsp(A),C) = k8_funct_2(u1_struct_0(A),u1_struct_0(k3_t_0topsp(A)),k4_t_0topsp(A),D) )
                     => r2_hidden(D,B) ) ) ) ) ) ) )).

fof(t11_t_0topsp,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,k1_zfmisc_1(u1_struct_0(A)))
               => ( r2_hidden(C,k2_t_0topsp(A))
                 => ( r1_xboole_0(C,B)
                    | r1_tarski(C,B) ) ) ) ) ) ) )).

fof(t12_t_0topsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => v1_t_0topsp(k4_t_0topsp(A),A,k3_t_0topsp(A)) ) )).

fof(d7_t_0topsp,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ( v2_t_0topsp(A)
      <=> ( v3_struct_0(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)))
                         => ~ ( v3_pre_topc(D,A)
                              & ( ( r2_hidden(B,D)
                                  & ~ r2_hidden(C,D) )
                                | ( r2_hidden(C,D)
                                  & ~ r2_hidden(B,D) ) ) ) ) ) ) ) ) ) ) )).

fof(t13_t_0topsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( ~ v3_struct_0(k3_t_0topsp(A))
        & v2_pre_topc(k3_t_0topsp(A))
        & v2_t_0topsp(k3_t_0topsp(A))
        & l1_pre_topc(k3_t_0topsp(A)) ) ) )).

fof(t14_t_0topsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(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(k3_t_0topsp(B)),u1_struct_0(k3_t_0topsp(A)))
                & m2_relset_1(C,u1_struct_0(k3_t_0topsp(B)),u1_struct_0(k3_t_0topsp(A)))
                & v3_tops_2(C,k3_t_0topsp(B),k3_t_0topsp(A))
                & r1_rfinseq(k4_t_0topsp(A),k7_funct_2(u1_struct_0(B),u1_struct_0(k3_t_0topsp(B)),u1_struct_0(k3_t_0topsp(A)),k4_t_0topsp(B),C)) )
           => r1_t_0topsp(A,B) ) ) ) )).

fof(t15_t_0topsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & v2_t_0topsp(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & v5_pre_topc(C,A,B)
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(A))
                     => ( r2_hidden(k8_borsuk_1(A,A,D,E),k1_t_0topsp(A))
                       => k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D) = k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E) ) ) ) ) ) ) )).

fof(t16_t_0topsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & v2_t_0topsp(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & v5_pre_topc(C,A,B)
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => k4_pre_topc(A,B,C,k6_eqrel_1(u1_struct_0(A),k1_t_0topsp(A),D)) = k1_struct_0(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D)) ) ) ) ) )).

fof(t17_t_0topsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & v2_t_0topsp(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & v5_pre_topc(C,A,B)
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ? [D] :
                  ( v1_funct_1(D)
                  & v1_funct_2(D,u1_struct_0(k3_t_0topsp(A)),u1_struct_0(B))
                  & v5_pre_topc(D,k3_t_0topsp(A),B)
                  & m2_relset_1(D,u1_struct_0(k3_t_0topsp(A)),u1_struct_0(B))
                  & C = k4_borsuk_1(A,k3_t_0topsp(A),B,k4_t_0topsp(A),D) ) ) ) ) )).

fof(dt_k1_t_0topsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A) )
     => ( v3_relat_2(k1_t_0topsp(A))
        & v8_relat_2(k1_t_0topsp(A))
        & v1_partfun1(k1_t_0topsp(A),u1_struct_0(A),u1_struct_0(A))
        & m2_relset_1(k1_t_0topsp(A),u1_struct_0(A),u1_struct_0(A)) ) ) )).

fof(dt_k2_t_0topsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A) )
     => ( ~ v1_xboole_0(k2_t_0topsp(A))
        & m1_eqrel_1(k2_t_0topsp(A),u1_struct_0(A)) ) ) )).

fof(dt_k3_t_0topsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v2_pre_topc(k3_t_0topsp(A))
        & l1_pre_topc(k3_t_0topsp(A)) ) ) )).

fof(dt_k4_t_0topsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v1_funct_1(k4_t_0topsp(A))
        & v1_funct_2(k4_t_0topsp(A),u1_struct_0(A),u1_struct_0(k3_t_0topsp(A)))
        & v5_pre_topc(k4_t_0topsp(A),A,k3_t_0topsp(A))
        & m2_relset_1(k4_t_0topsp(A),u1_struct_0(A),u1_struct_0(k3_t_0topsp(A))) ) ) )).

fof(t5_t_0topsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( u1_struct_0(k3_t_0topsp(A)) = k2_t_0topsp(A)
        & u1_pre_topc(k3_t_0topsp(A)) = a_1_0_t_0topsp(A) ) ) )).

fof(fraenkel_a_1_0_t_0topsp,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & v2_pre_topc(B)
        & l1_pre_topc(B) )
     => ( r2_hidden(A,a_1_0_t_0topsp(B))
      <=> ? [C] :
            ( m1_subset_1(C,k1_zfmisc_1(k2_t_0topsp(B)))
            & A = C
            & r2_hidden(k3_tarski(C),u1_pre_topc(B)) ) ) ) )).
%------------------------------------------------------------------------------