SET007 Axioms: SET007+541.ax


%------------------------------------------------------------------------------
% File     : SET007+541 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : On Tone Reflex of 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    : t_1topsp [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   40 (   3 unit)
%            Number of atoms       :  264 (  21 equality)
%            Maximal formula depth :   15 (   8 average)
%            Number of connectives :  282 (  58 ~  ;   0  |; 133  &)
%                                         (   9 <=>;  82 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   23 (   1 propositional; 0-3 arity)
%            Number of functors    :   23 (   0 constant; 1-5 arity)
%            Number of variables   :  102 (   0 singleton;  95 !;   7 ?)
%            Maximal term depth    :    5 (   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(rc1_t_1topsp,axiom,(
    ! [A] :
    ? [B] :
      ( m1_t_1topsp(B,A)
      & v1_t_1topsp(B,A) ) )).

fof(rc2_t_1topsp,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ? [B] :
          ( m1_eqrel_1(B,A)
          & ~ v1_xboole_0(B)
          & v1_setfam_1(B) ) ) )).

fof(rc3_t_1topsp,axiom,(
    ! [A] :
    ? [B] :
      ( m1_t_1topsp(B,A)
      & ~ v1_xboole_0(B)
      & v1_t_1topsp(B,A) ) )).

fof(fc1_t_1topsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( ~ v3_struct_0(k4_t_1topsp(A))
        & v1_pre_topc(k4_t_1topsp(A))
        & v2_pre_topc(k4_t_1topsp(A)) ) ) )).

fof(fc2_t_1topsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( ~ v3_struct_0(k4_t_1topsp(A))
        & v1_pre_topc(k4_t_1topsp(A))
        & v2_pre_topc(k4_t_1topsp(A))
        & v1_urysohn1(k4_t_1topsp(A)) ) ) )).

fof(t1_t_1topsp,axiom,(
    $true )).

fof(t2_t_1topsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_eqrel_1(B,u1_struct_0(A)) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k16_borsuk_1(A,B))))
             => k5_pre_topc(A,k16_borsuk_1(A,B),k17_borsuk_1(A,B),C) = k3_tarski(C) ) ) ) )).

fof(t3_t_1topsp,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(B))
             => k4_xboole_0(k5_setfam_1(A,B),k3_tarski(C)) = k3_tarski(k4_xboole_0(B,C)) ) ) ) )).

fof(t4_t_1topsp,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => ( r2_hidden(B,C)
               => k3_tarski(k4_xboole_0(C,k1_tarski(B))) = k4_xboole_0(A,B) ) ) ) ) )).

fof(t5_t_1topsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_eqrel_1(B,u1_struct_0(A)) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k16_borsuk_1(A,B))))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => ( D = k3_tarski(C)
                   => ( v4_pre_topc(C,k16_borsuk_1(A,B))
                    <=> v4_pre_topc(D,A) ) ) ) ) ) ) )).

fof(d1_t_1topsp,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(A))
                 => ( D = k1_t_1topsp(A,B,C)
                  <=> ( r2_hidden(B,D)
                      & r2_hidden(D,C) ) ) ) ) ) ) )).

fof(t6_t_1topsp,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => ( ! [D] :
                    ( m1_subset_1(D,A)
                   => k1_t_1topsp(A,D,B) = k1_t_1topsp(A,D,C) )
               => B = C ) ) ) ) )).

fof(t7_t_1topsp,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => m1_eqrel_1(k1_tarski(A),A) ) )).

fof(d2_t_1topsp,axiom,(
    ! [A,B] :
      ( m1_t_1topsp(B,A)
    <=> r1_tarski(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) )).

fof(d3_t_1topsp,axiom,(
    ! [A,B] :
      ( m1_t_1topsp(B,A)
     => ( v1_t_1topsp(B,A)
      <=> ! [C] :
            ( r2_hidden(C,B)
           => m1_eqrel_1(C,A) ) ) ) )).

fof(t8_t_1topsp,axiom,(
    ! [A,B] :
      ( m1_eqrel_1(B,A)
     => ( v1_t_1topsp(k1_tarski(B),A)
        & m1_t_1topsp(k1_tarski(B),A) ) ) )).

fof(t9_t_1topsp,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( m1_subset_1(D,A)
                 => ( ~ r1_xboole_0(k1_t_1topsp(A,C,B),k1_t_1topsp(A,D,B))
                   => k1_t_1topsp(A,C,B) = k1_t_1topsp(A,D,B) ) ) ) ) ) )).

fof(t10_t_1topsp,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( m1_eqrel_1(C,B)
         => ~ ( r2_hidden(A,C)
              & ! [D] :
                  ( m1_subset_1(D,B)
                 => A != k1_t_1topsp(B,D,C) ) ) ) ) )).

fof(d6_t_1topsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => k4_t_1topsp(A) = k16_borsuk_1(A,k2_t_1topsp(u1_struct_0(A),k3_t_1topsp(A))) ) )).

fof(t12_t_1topsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => v1_urysohn1(k4_t_1topsp(A)) ) )).

fof(d7_t_1topsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => k5_t_1topsp(A) = k17_borsuk_1(A,k2_t_1topsp(u1_struct_0(A),k3_t_1topsp(A))) ) )).

fof(t14_t_1topsp,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(A),u1_struct_0(B))
                & v5_pre_topc(C,A,B)
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( v1_urysohn1(B)
               => ! [D,E] :
                    ( m1_subset_1(E,u1_struct_0(A))
                   => ( D = k1_t_1topsp(u1_struct_0(A),E,k2_t_1topsp(u1_struct_0(A),k3_t_1topsp(A)))
                     => r1_tarski(D,k5_pre_topc(A,B,C,k1_struct_0(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E)))) ) ) ) ) ) ) )).

fof(t15_t_1topsp,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(A),u1_struct_0(B))
                & v5_pre_topc(C,A,B)
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( v1_urysohn1(B)
               => ! [D] :
                    ~ ( r2_hidden(D,u1_struct_0(k4_t_1topsp(A)))
                      & ! [E] :
                          ( m1_subset_1(E,u1_struct_0(B))
                         => ~ ( r2_hidden(E,k2_relat_1(C))
                              & r1_tarski(D,k5_pre_topc(A,B,C,k1_struct_0(B,E))) ) ) ) ) ) ) ) )).

fof(t16_t_1topsp,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(A),u1_struct_0(B))
                & v5_pre_topc(C,A,B)
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ~ ( v1_urysohn1(B)
                  & ! [D] :
                      ( ( v1_funct_1(D)
                        & v1_funct_2(D,u1_struct_0(k4_t_1topsp(A)),u1_struct_0(B))
                        & v5_pre_topc(D,k4_t_1topsp(A),B)
                        & m2_relset_1(D,u1_struct_0(k4_t_1topsp(A)),u1_struct_0(B)) )
                     => C != k4_borsuk_1(A,k4_t_1topsp(A),B,k5_t_1topsp(A),D) ) ) ) ) ) )).

fof(d8_t_1topsp,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(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(k4_t_1topsp(A)),u1_struct_0(k4_t_1topsp(B)))
                    & v5_pre_topc(D,k4_t_1topsp(A),k4_t_1topsp(B))
                    & m2_relset_1(D,u1_struct_0(k4_t_1topsp(A)),u1_struct_0(k4_t_1topsp(B))) )
                 => ( D = k6_t_1topsp(A,B,C)
                  <=> k4_borsuk_1(A,B,k4_t_1topsp(B),C,k5_t_1topsp(B)) = k4_borsuk_1(A,k4_t_1topsp(A),k4_t_1topsp(B),k5_t_1topsp(A),D) ) ) ) ) ) )).

fof(dt_m1_t_1topsp,axiom,(
    $true )).

fof(existence_m1_t_1topsp,axiom,(
    ! [A] :
    ? [B] : m1_t_1topsp(B,A) )).

fof(dt_k1_t_1topsp,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & m1_eqrel_1(C,A) )
     => m1_subset_1(k1_t_1topsp(A,B,C),k1_zfmisc_1(A)) ) )).

fof(dt_k2_t_1topsp,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & v1_t_1topsp(B,A)
        & m1_t_1topsp(B,A) )
     => ( ~ v1_xboole_0(k2_t_1topsp(A,B))
        & m1_eqrel_1(k2_t_1topsp(A,B),A) ) ) )).

fof(dt_k3_t_1topsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( ~ v1_xboole_0(k3_t_1topsp(A))
        & v1_t_1topsp(k3_t_1topsp(A),u1_struct_0(A))
        & m1_t_1topsp(k3_t_1topsp(A),u1_struct_0(A)) ) ) )).

fof(dt_k4_t_1topsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v2_pre_topc(k4_t_1topsp(A))
        & l1_pre_topc(k4_t_1topsp(A)) ) ) )).

fof(dt_k5_t_1topsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v1_funct_1(k5_t_1topsp(A))
        & v1_funct_2(k5_t_1topsp(A),u1_struct_0(A),u1_struct_0(k4_t_1topsp(A)))
        & v5_pre_topc(k5_t_1topsp(A),A,k4_t_1topsp(A))
        & m2_relset_1(k5_t_1topsp(A),u1_struct_0(A),u1_struct_0(k4_t_1topsp(A))) ) ) )).

fof(dt_k6_t_1topsp,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & ~ v3_struct_0(B)
        & v2_pre_topc(B)
        & l1_pre_topc(B)
        & v1_funct_1(C)
        & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
        & v5_pre_topc(C,A,B)
        & m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
     => ( v1_funct_1(k6_t_1topsp(A,B,C))
        & v1_funct_2(k6_t_1topsp(A,B,C),u1_struct_0(k4_t_1topsp(A)),u1_struct_0(k4_t_1topsp(B)))
        & v5_pre_topc(k6_t_1topsp(A,B,C),k4_t_1topsp(A),k4_t_1topsp(B))
        & m2_relset_1(k6_t_1topsp(A,B,C),u1_struct_0(k4_t_1topsp(A)),u1_struct_0(k4_t_1topsp(B))) ) ) )).

fof(d4_t_1topsp,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_t_1topsp(B,A)
            & m1_t_1topsp(B,A) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_eqrel_1(C,A) )
             => ( C = k2_t_1topsp(A,B)
              <=> ! [D] :
                    ( m1_subset_1(D,A)
                   => k1_t_1topsp(A,D,C) = k1_setfam_1(a_3_0_t_1topsp(A,B,D)) ) ) ) ) ) )).

fof(t11_t_1topsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v1_t_1topsp(a_1_0_t_1topsp(A),u1_struct_0(A))
        & m1_t_1topsp(a_1_0_t_1topsp(A),u1_struct_0(A)) ) ) )).

fof(d5_t_1topsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => k3_t_1topsp(A) = a_1_0_t_1topsp(A) ) )).

fof(t13_t_1topsp,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(A),u1_struct_0(B))
                & v5_pre_topc(C,A,B)
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( v1_urysohn1(B)
               => ( m1_eqrel_1(a_3_1_t_1topsp(A,B,C),u1_struct_0(A))
                  & ! [D] :
                      ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                     => ( r2_hidden(D,a_3_1_t_1topsp(A,B,C))
                       => v4_pre_topc(D,A) ) ) ) ) ) ) ) )).

fof(fraenkel_a_3_0_t_1topsp,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & v1_t_1topsp(C,B)
        & m1_t_1topsp(C,B)
        & m1_subset_1(D,B) )
     => ( r2_hidden(A,a_3_0_t_1topsp(B,C,D))
      <=> ? [E] :
            ( m1_eqrel_1(E,B)
            & A = k1_t_1topsp(B,D,E)
            & r2_hidden(E,C) ) ) ) )).

fof(fraenkel_a_1_0_t_1topsp,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & v2_pre_topc(B)
        & l1_pre_topc(B) )
     => ( r2_hidden(A,a_1_0_t_1topsp(B))
      <=> ? [C] :
            ( m1_eqrel_1(C,u1_struct_0(B))
            & A = C
            & v2_tops_2(C,B) ) ) ) )).

fof(fraenkel_a_3_1_t_1topsp,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & v2_pre_topc(B)
        & l1_pre_topc(B)
        & ~ v3_struct_0(C)
        & v2_pre_topc(C)
        & l1_pre_topc(C)
        & v1_funct_1(D)
        & v1_funct_2(D,u1_struct_0(B),u1_struct_0(C))
        & v5_pre_topc(D,B,C)
        & m2_relset_1(D,u1_struct_0(B),u1_struct_0(C)) )
     => ( r2_hidden(A,a_3_1_t_1topsp(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(C))
            & A = k5_pre_topc(B,C,D,k1_struct_0(C,E))
            & r2_hidden(E,k2_relat_1(D)) ) ) ) )).
%------------------------------------------------------------------------------