SET007 Axioms: SET007+872.ax


%------------------------------------------------------------------------------
% File     : SET007+872 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : On Some Points of a Simple Closed Curve. Part II
% 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    : jordan22 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   17 (   0 unit)
%            Number of atoms       :   70 (   0 equality)
%            Maximal formula depth :    8 (   5 average)
%            Number of connectives :   55 (   2 ~  ;   1  |;  31  &)
%                                         (   0 <=>;  21 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   11 (   0 propositional; 1-3 arity)
%            Number of functors    :   21 (   4 constant; 0-3 arity)
%            Number of variables   :   21 (   0 singleton;  21 !;   0 ?)
%            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_jordan22,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ( ~ v1_xboole_0(k8_jordan6(A))
        & v4_pre_topc(k8_jordan6(A),k15_euclid(np__2))
        & v2_connsp_1(k8_jordan6(A),k15_euclid(np__2))
        & v1_jordan2c(k8_jordan6(A),np__2)
        & v6_compts_1(k8_jordan6(A),k15_euclid(np__2))
        & v1_jordan21(k8_jordan6(A)) ) ) )).

fof(fc2_jordan22,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ( ~ v1_xboole_0(k9_jordan6(A))
        & v4_pre_topc(k9_jordan6(A),k15_euclid(np__2))
        & v2_connsp_1(k9_jordan6(A),k15_euclid(np__2))
        & v1_jordan2c(k9_jordan6(A),np__2)
        & v6_compts_1(k9_jordan6(A),k15_euclid(np__2))
        & v1_jordan21(k9_jordan6(A)) ) ) )).

fof(fc3_jordan22,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => v6_compts_1(k3_jordan19(A),k15_euclid(np__2)) ) )).

fof(fc4_jordan22,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => v6_compts_1(k4_jordan19(A),k15_euclid(np__2)) ) )).

fof(t1_jordan22,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => r1_tarski(k7_kurato_2(u1_struct_0(k15_euclid(np__2)),k1_jordan19(A),B),k6_pre_topc(k15_euclid(np__2),k3_goboard9(k1_jordan9(A,np__0)))) ) ) )).

fof(t2_jordan22,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => r1_tarski(k7_kurato_2(u1_struct_0(k15_euclid(np__2)),k2_jordan19(A),B),k6_pre_topc(k15_euclid(np__2),k3_goboard9(k1_jordan9(A,np__0)))) ) ) )).

fof(t3_jordan22,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( v6_compts_1(k7_kurato_2(u1_struct_0(k15_euclid(np__2)),k1_jordan19(A),B),k15_euclid(np__2))
            & v2_connsp_1(k7_kurato_2(u1_struct_0(k15_euclid(np__2)),k1_jordan19(A),B),k15_euclid(np__2)) ) ) ) )).

fof(t4_jordan22,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( v6_compts_1(k7_kurato_2(u1_struct_0(k15_euclid(np__2)),k2_jordan19(A),B),k15_euclid(np__2))
            & v2_connsp_1(k7_kurato_2(u1_struct_0(k15_euclid(np__2)),k2_jordan19(A),B),k15_euclid(np__2)) ) ) ) )).

fof(t5_jordan22,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => r2_hidden(k30_pscomp_1(A),k3_jordan19(A)) ) )).

fof(t6_jordan22,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => r2_hidden(k34_pscomp_1(A),k3_jordan19(A)) ) )).

fof(t7_jordan22,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => r2_hidden(k30_pscomp_1(A),k4_jordan19(A)) ) )).

fof(t8_jordan22,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => r2_hidden(k34_pscomp_1(A),k4_jordan19(A)) ) )).

fof(t9_jordan22,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => r2_hidden(k1_jordan21(A),k3_jordan19(A)) ) )).

fof(t10_jordan22,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => r2_hidden(k2_jordan21(A),k4_jordan19(A)) ) )).

fof(t11_jordan22,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => r1_tarski(k3_jordan19(A),A) ) )).

fof(t12_jordan22,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => r1_tarski(k4_jordan19(A),A) ) )).

fof(t13_jordan22,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ( ( r2_hidden(k2_jordan21(A),k9_jordan6(A))
          & r2_hidden(k1_jordan21(A),k8_jordan6(A)) )
        | ( r2_hidden(k1_jordan21(A),k9_jordan6(A))
          & r2_hidden(k2_jordan21(A),k8_jordan6(A)) ) ) ) )).
%------------------------------------------------------------------------------