SET007 Axioms: SET007+326.ax


%------------------------------------------------------------------------------
% File     : SET007+326 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Topological Space E^2_T.  Simple Closed Curves
% 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    : topreal2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   10 (   1 unit)
%            Number of atoms       :   89 (  13 equality)
%            Maximal formula depth :   21 (  11 average)
%            Number of connectives :  103 (  24 ~  ;   0  |;  51  &)
%                                         (   3 <=>;  25 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   11 (   0 propositional; 1-4 arity)
%            Number of functors    :   10 (   2 constant; 0-3 arity)
%            Number of variables   :   30 (   0 singleton;  22 !;   8 ?)
%            Maximal term depth    :    4 (   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(fc1_topreal2,axiom,
    ( ~ v1_xboole_0(k2_topreal1)
    & v1_topreal2(k2_topreal1) )).

fof(rc1_topreal2,axiom,(
    ? [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
      & v1_topreal2(A) ) )).

fof(cc1_topreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => ( v1_topreal2(A)
       => ( ~ v1_xboole_0(A)
          & v6_compts_1(A,k15_euclid(np__2)) ) ) ) )).

fof(t1_topreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
         => ~ ( A != B
              & r2_hidden(A,k2_topreal1)
              & r2_hidden(B,k2_topreal1)
              & ! [C] :
                  ( ( ~ v1_xboole_0(C)
                    & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
                 => ! [D] :
                      ( ( ~ v1_xboole_0(D)
                        & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
                     => ~ ( r1_topreal1(k15_euclid(np__2),A,B,C)
                          & r1_topreal1(k15_euclid(np__2),A,B,D)
                          & k2_topreal1 = k4_subset_1(u1_struct_0(k15_euclid(np__2)),C,D)
                          & k5_subset_1(u1_struct_0(k15_euclid(np__2)),C,D) = k2_struct_0(k15_euclid(np__2),A,B) ) ) ) ) ) ) )).

fof(t2_topreal2,axiom,(
    v6_compts_1(k2_topreal1,k15_euclid(np__2)) )).

fof(t3_topreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(k3_pre_topc(k15_euclid(np__2),C)),u1_struct_0(k3_pre_topc(k15_euclid(np__2),D)))
                        & m2_relset_1(E,u1_struct_0(k3_pre_topc(k15_euclid(np__2),C)),u1_struct_0(k3_pre_topc(k15_euclid(np__2),D))) )
                     => ( ( v3_tops_2(E,k3_pre_topc(k15_euclid(np__2),C),k3_pre_topc(k15_euclid(np__2),D))
                          & r1_topreal1(k15_euclid(np__2),A,B,C) )
                       => ! [F] :
                            ( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
                           => ! [G] :
                                ( m1_subset_1(G,u1_struct_0(k15_euclid(np__2)))
                               => ( ( F = k1_funct_1(E,A)
                                    & G = k1_funct_1(E,B) )
                                 => r1_topreal1(k15_euclid(np__2),F,G,D) ) ) ) ) ) ) ) ) ) )).

fof(d1_topreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => ( v1_topreal2(A)
      <=> ? [B] :
            ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(k3_pre_topc(k15_euclid(np__2),k2_topreal1)),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A)))
            & m2_relset_1(B,u1_struct_0(k3_pre_topc(k15_euclid(np__2),k2_topreal1)),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A)))
            & v3_tops_2(B,k3_pre_topc(k15_euclid(np__2),k2_topreal1),k3_pre_topc(k15_euclid(np__2),A)) ) ) ) )).

fof(t4_topreal2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ~ ( v1_topreal2(A)
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
                 => ~ ( B != C
                      & r2_hidden(B,A)
                      & r2_hidden(C,A) ) ) ) ) ) )).

fof(t5_topreal2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ( v1_topreal2(A)
      <=> ( ? [B] :
              ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
              & ? [C] :
                  ( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
                  & B != C
                  & r2_hidden(B,A)
                  & r2_hidden(C,A) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
                 => ~ ( B != C
                      & r2_hidden(B,A)
                      & r2_hidden(C,A)
                      & ! [D] :
                          ( ( ~ v1_xboole_0(D)
                            & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
                         => ! [E] :
                              ( ( ~ v1_xboole_0(E)
                                & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
                             => ~ ( r1_topreal1(k15_euclid(np__2),B,C,D)
                                  & r1_topreal1(k15_euclid(np__2),B,C,E)
                                  & A = k4_subset_1(u1_struct_0(k15_euclid(np__2)),D,E)
                                  & k5_subset_1(u1_struct_0(k15_euclid(np__2)),D,E) = k2_struct_0(k15_euclid(np__2),B,C) ) ) ) ) ) ) ) ) ) )).

fof(t6_topreal2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ( v1_topreal2(A)
      <=> ? [B] :
            ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
            & ? [C] :
                ( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
                & ? [D] :
                    ( ~ v1_xboole_0(D)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                    & ? [E] :
                        ( ~ v1_xboole_0(E)
                        & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                        & B != C
                        & r2_hidden(B,A)
                        & r2_hidden(C,A)
                        & r1_topreal1(k15_euclid(np__2),B,C,D)
                        & r1_topreal1(k15_euclid(np__2),B,C,E)
                        & A = k4_subset_1(u1_struct_0(k15_euclid(np__2)),D,E)
                        & k5_subset_1(u1_struct_0(k15_euclid(np__2)),D,E) = k2_struct_0(k15_euclid(np__2),B,C) ) ) ) ) ) ) )).
%------------------------------------------------------------------------------