SET007 Axioms: SET007+669.ax


%------------------------------------------------------------------------------
% File     : SET007+669 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Some Lemmas for the Jordan Curve Theorem
% 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    : jct_misc [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   35 (   5 unit)
%            Number of atoms       :  244 (  18 equality)
%            Maximal formula depth :   24 (   9 average)
%            Number of connectives :  249 (  40 ~  ;   0  |; 109  &)
%                                         (   6 <=>;  94 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   26 (   1 propositional; 0-3 arity)
%            Number of functors    :   44 (  10 constant; 0-4 arity)
%            Number of variables   :  115 (   0 singleton; 104 !;  11 ?)
%            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(t1_jct_misc,axiom,(
    $true )).

fof(t2_jct_misc,axiom,(
    $true )).

fof(t3_jct_misc,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( ( r1_tarski(A,k1_relat_1(C))
          & r1_tarski(k9_relat_1(C,A),B) )
       => r1_tarski(A,k10_relat_1(C,B)) ) ) )).

fof(t4_jct_misc,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C] :
          ( r1_xboole_0(B,C)
         => r1_xboole_0(k10_relat_1(A,B),k10_relat_1(A,C)) ) ) )).

fof(t5_jct_misc,axiom,(
    ! [A,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(B))
         => ( ( B = k1_xboole_0
             => A = k1_xboole_0 )
           => k3_subset_1(A,k3_funct_2(A,B,C,D)) = k3_funct_2(A,B,C,k3_subset_1(B,D)) ) ) ) )).

fof(t6_jct_misc,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),B)
                & m2_relset_1(C,u1_struct_0(A),B) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(B))
                 => k3_subset_1(u1_struct_0(A),k3_funct_2(u1_struct_0(A),B,C,D)) = k3_funct_2(u1_struct_0(A),B,C,k3_subset_1(B,D)) ) ) ) ) )).

fof(t7_jct_misc,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(A,B)
           => k5_binarith(B,k5_binarith(B,A)) = A ) ) ) )).

fof(t8_jct_misc,axiom,(
    $true )).

fof(t9_jct_misc,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( ( r2_hidden(A,k1_rcomp_1(C,D))
                      & r2_hidden(B,k1_rcomp_1(C,D)) )
                   => r2_hidden(k7_xcmplx_0(k2_xcmplx_0(A,B),np__2),k1_rcomp_1(C,D)) ) ) ) ) ) )).

fof(t10_jct_misc,axiom,(
    $true )).

fof(t11_jct_misc,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => r1_xreal_0(k18_complex1(k5_real_1(k18_complex1(k6_xcmplx_0(A,B)),k18_complex1(k6_xcmplx_0(C,D)))),k3_real_1(k18_complex1(k6_xcmplx_0(A,C)),k18_complex1(k6_xcmplx_0(B,D)))) ) ) ) ) )).

fof(t12_jct_misc,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ~ ( r2_hidden(A,k2_rcomp_1(B,C))
                  & r1_xreal_0(k4_square_1(k18_complex1(B),k18_complex1(C)),k18_complex1(A)) ) ) ) ) )).

fof(d1_jct_misc,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,A,k2_zfmisc_1(B,C))
                    & m2_relset_1(D,A,k2_zfmisc_1(B,C)) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,A,B)
                        & m2_relset_1(E,A,B) )
                     => ( E = k1_jct_misc(A,B,C,D)
                      <=> ! [F] :
                            ( m1_subset_1(F,A)
                           => k8_funct_2(A,B,E,F) = k1_mcart_1(k8_funct_2(A,k2_zfmisc_1(B,C),D,F)) ) ) ) ) ) ) ) )).

fof(d2_jct_misc,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,A,k2_zfmisc_1(B,C))
                    & m2_relset_1(D,A,k2_zfmisc_1(B,C)) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,A,C)
                        & m2_relset_1(E,A,C) )
                     => ( E = k2_jct_misc(A,B,C,D)
                      <=> ! [F] :
                            ( m1_subset_1(F,A)
                           => k8_funct_2(A,C,E,F) = k2_mcart_1(k8_funct_2(A,k2_zfmisc_1(B,C),D,F)) ) ) ) ) ) ) ) )).

fof(t13_jct_misc,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] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B))))
             => ( ! [D] :
                    ( m1_subset_1(D,u1_struct_0(k6_borsuk_1(A,B)))
                   => ~ ( r2_hidden(D,C)
                        & ! [E] :
                            ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                           => ! [F] :
                                ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(B)))
                               => ~ ( v3_pre_topc(E,A)
                                    & v3_pre_topc(F,B)
                                    & r2_hidden(D,k7_borsuk_1(A,B,E,F))
                                    & r1_tarski(k7_borsuk_1(A,B,E,F),C) ) ) ) ) )
               => v3_pre_topc(C,k6_borsuk_1(A,B)) ) ) ) ) )).

fof(t14_jct_misc,axiom,(
    ! [A] :
      ( ( v1_rcomp_1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( ( v1_rcomp_1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => v1_rcomp_1(k5_subset_1(k1_numbers,A,B)) ) ) )).

fof(d3_jct_misc,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ( v1_jct_misc(A)
      <=> ! [B] :
            ( v1_xreal_0(B)
           => ! [C] :
                ( v1_xreal_0(C)
               => ( ( r2_hidden(B,A)
                    & r2_hidden(C,A) )
                 => r1_tarski(k1_rcomp_1(B,C),A) ) ) ) ) ) )).

fof(t15_jct_misc,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(A),k1_numbers)
            & v9_pscomp_1(B,A)
            & m2_relset_1(B,u1_struct_0(A),k1_numbers) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( v2_connsp_1(C,A)
               => v1_jct_misc(k2_funct_2(u1_struct_0(A),k1_numbers,B,C)) ) ) ) ) )).

fof(t16_jct_misc,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( ( r2_hidden(C,A)
                      & r2_hidden(D,B) )
                   => r1_xreal_0(k3_jct_misc(A,B),k18_complex1(k6_xcmplx_0(C,D))) ) ) ) ) ) )).

fof(t17_jct_misc,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(k1_numbers)) )
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & m1_subset_1(D,k1_zfmisc_1(k1_numbers)) )
                 => ( ( r1_tarski(C,A)
                      & r1_tarski(D,B) )
                   => r1_xreal_0(k3_jct_misc(A,B),k3_jct_misc(C,D)) ) ) ) ) ) )).

fof(t18_jct_misc,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_rcomp_1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_rcomp_1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ? [C] :
              ( v1_xreal_0(C)
              & ? [D] :
                  ( v1_xreal_0(D)
                  & r2_hidden(C,A)
                  & r2_hidden(D,B)
                  & k3_jct_misc(A,B) = k18_complex1(k6_xcmplx_0(C,D)) ) ) ) ) )).

fof(t19_jct_misc,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_rcomp_1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_rcomp_1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => r1_xreal_0(np__0,k3_jct_misc(A,B)) ) ) )).

fof(t20_jct_misc,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_rcomp_1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_rcomp_1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ~ ( r1_subset_1(A,B)
              & r1_xreal_0(k3_jct_misc(A,B),np__0) ) ) ) )).

fof(t21_jct_misc,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( ( v1_rcomp_1(C)
                & m1_subset_1(C,k1_zfmisc_1(k1_numbers)) )
             => ! [D] :
                  ( ( v1_rcomp_1(D)
                    & m1_subset_1(D,k1_zfmisc_1(k1_numbers)) )
                 => ( ( r1_xboole_0(C,D)
                      & r1_tarski(C,k1_rcomp_1(A,B))
                      & r1_tarski(D,k1_rcomp_1(A,B)) )
                   => ! [E] :
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,k5_numbers,k1_zfmisc_1(k1_numbers))
                          & m2_relset_1(E,k5_numbers,k1_zfmisc_1(k1_numbers)) )
                       => ~ ( ! [F] :
                                ( m2_subset_1(F,k1_numbers,k5_numbers)
                               => ( v1_jct_misc(k8_funct_2(k5_numbers,k1_zfmisc_1(k1_numbers),E,F))
                                  & ~ r1_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(k1_numbers),E,F),C)
                                  & ~ r1_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(k1_numbers),E,F),D) ) )
                            & ! [F] :
                                ( v1_xreal_0(F)
                               => ~ ( r2_hidden(F,k1_rcomp_1(A,B))
                                    & ~ r2_hidden(F,k4_subset_1(k1_numbers,C,D))
                                    & ! [G] :
                                        ( m2_subset_1(G,k1_numbers,k5_numbers)
                                       => ? [H] :
                                            ( m2_subset_1(H,k1_numbers,k5_numbers)
                                            & r1_xreal_0(G,H)
                                            & r2_hidden(F,k8_funct_2(k5_numbers,k1_zfmisc_1(k1_numbers),E,H)) ) ) ) ) ) ) ) ) ) ) ) )).

fof(s2_jct_misc,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s2_jct_misc)
       => ? [B] :
            ( m1_subset_1(B,f2_s2_jct_misc)
            & ? [C] :
                ( m1_subset_1(C,f3_s2_jct_misc)
                & p1_s2_jct_misc(A,B,C) ) ) )
   => ? [A] :
        ( v1_funct_1(A)
        & v1_funct_2(A,f1_s2_jct_misc,f2_s2_jct_misc)
        & m2_relset_1(A,f1_s2_jct_misc,f2_s2_jct_misc)
        & ? [B] :
            ( v1_funct_1(B)
            & v1_funct_2(B,f1_s2_jct_misc,f3_s2_jct_misc)
            & m2_relset_1(B,f1_s2_jct_misc,f3_s2_jct_misc)
            & ! [C] :
                ( m1_subset_1(C,f1_s2_jct_misc)
               => p1_s2_jct_misc(C,k8_funct_2(f1_s2_jct_misc,f2_s2_jct_misc,A,C),k8_funct_2(f1_s2_jct_misc,f3_s2_jct_misc,B,C)) ) ) ) )).

fof(dt_k1_jct_misc,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & v1_funct_1(D)
        & v1_funct_2(D,A,k2_zfmisc_1(B,C))
        & m1_relset_1(D,A,k2_zfmisc_1(B,C)) )
     => ( v1_funct_1(k1_jct_misc(A,B,C,D))
        & v1_funct_2(k1_jct_misc(A,B,C,D),A,B)
        & m2_relset_1(k1_jct_misc(A,B,C,D),A,B) ) ) )).

fof(redefinition_k1_jct_misc,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & v1_funct_1(D)
        & v1_funct_2(D,A,k2_zfmisc_1(B,C))
        & m1_relset_1(D,A,k2_zfmisc_1(B,C)) )
     => k1_jct_misc(A,B,C,D) = k15_mcart_1(D) ) )).

fof(dt_k2_jct_misc,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & v1_funct_1(D)
        & v1_funct_2(D,A,k2_zfmisc_1(B,C))
        & m1_relset_1(D,A,k2_zfmisc_1(B,C)) )
     => ( v1_funct_1(k2_jct_misc(A,B,C,D))
        & v1_funct_2(k2_jct_misc(A,B,C,D),A,C)
        & m2_relset_1(k2_jct_misc(A,B,C,D),A,C) ) ) )).

fof(redefinition_k2_jct_misc,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & v1_funct_1(D)
        & v1_funct_2(D,A,k2_zfmisc_1(B,C))
        & m1_relset_1(D,A,k2_zfmisc_1(B,C)) )
     => k2_jct_misc(A,B,C,D) = k16_mcart_1(D) ) )).

fof(dt_k3_jct_misc,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
        & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
     => v1_xreal_0(k3_jct_misc(A,B)) ) )).

fof(commutativity_k3_jct_misc,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
        & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
     => k3_jct_misc(A,B) = k3_jct_misc(B,A) ) )).

fof(d4_jct_misc,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
         => ! [C] :
              ( v1_xreal_0(C)
             => ( C = k3_jct_misc(A,B)
              <=> ? [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(k1_numbers))
                    & D = a_2_0_jct_misc(A,B)
                    & C = k5_seq_4(D) ) ) ) ) ) )).

fof(s1_jct_misc,axiom,(
    ~ v1_xboole_0(a_0_0_jct_misc) )).

fof(fraenkel_a_2_0_jct_misc,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
        & m1_subset_1(C,k1_zfmisc_1(k1_numbers)) )
     => ( r2_hidden(A,a_2_0_jct_misc(B,C))
      <=> ? [D,E] :
            ( m1_subset_1(D,k1_numbers)
            & m1_subset_1(E,k1_numbers)
            & A = k18_complex1(k5_real_1(D,E))
            & r2_hidden(D,B)
            & r2_hidden(E,C) ) ) ) )).

fof(fraenkel_a_0_0_jct_misc,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_0_jct_misc)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s1_jct_misc)
          & A = f2_s1_jct_misc(B) ) ) )).
%------------------------------------------------------------------------------