SET007 Axioms: SET007+90.ax


%------------------------------------------------------------------------------
% File     : SET007+90 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Topological Properties of Subsets in Real Numbers
% 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    : rcomp_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   66 (  10 unit)
%            Number of atoms       :  310 (  16 equality)
%            Maximal formula depth :   21 (   7 average)
%            Number of connectives :  277 (  33 ~  ;   3  |;  83  &)
%                                         (  12 <=>; 146 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   25 (   1 propositional; 0-3 arity)
%            Number of functors    :   24 (   5 constant; 0-4 arity)
%            Number of variables   :  137 (   0 singleton; 129 !;   8 ?)
%            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_rcomp_1,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => ( v1_membered(k2_rcomp_1(A,B))
        & v2_membered(k2_rcomp_1(A,B))
        & v3_rcomp_1(k2_rcomp_1(A,B)) ) ) )).

fof(fc2_rcomp_1,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => ( v1_membered(k1_rcomp_1(A,B))
        & v2_membered(k1_rcomp_1(A,B))
        & v2_rcomp_1(k1_rcomp_1(A,B)) ) ) )).

fof(rc1_rcomp_1,axiom,(
    ? [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
      & v1_membered(A)
      & v2_membered(A)
      & v3_rcomp_1(A) ) )).

fof(cc1_rcomp_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m1_rcomp_1(B,A)
         => ( v1_membered(B)
            & v2_membered(B)
            & v3_rcomp_1(B) ) ) ) )).

fof(t1_rcomp_1,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)
               => ( r2_hidden(C,A)
                 => r2_hidden(C,B) ) )
           => r1_tarski(A,B) ) ) ) )).

fof(t2_rcomp_1,axiom,(
    $true )).

fof(t3_rcomp_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
         => ( ( r1_tarski(A,B)
              & v2_seq_4(B) )
           => v2_seq_4(A) ) ) ) )).

fof(t4_rcomp_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
         => ( ( r1_tarski(A,B)
              & v1_seq_4(B) )
           => v1_seq_4(A) ) ) ) )).

fof(t5_rcomp_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
         => ( ( r1_tarski(A,B)
              & v3_seq_4(B) )
           => v3_seq_4(A) ) ) ) )).

fof(t6_rcomp_1,axiom,(
    $true )).

fof(t7_rcomp_1,axiom,(
    $true )).

fof(t8_rcomp_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( r2_hidden(A,k2_rcomp_1(k6_xcmplx_0(B,C),k2_xcmplx_0(B,C)))
              <=> ~ r1_xreal_0(C,k18_complex1(k6_xcmplx_0(A,B))) ) ) ) ) )).

fof(t9_rcomp_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( r2_hidden(A,k1_rcomp_1(B,C))
              <=> r1_xreal_0(k18_complex1(k6_xcmplx_0(k2_xcmplx_0(B,C),k3_xcmplx_0(np__2,A))),k6_xcmplx_0(C,B)) ) ) ) ) )).

fof(t10_rcomp_1,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(k6_xcmplx_0(C,B),k18_complex1(k6_xcmplx_0(k2_xcmplx_0(B,C),k3_xcmplx_0(np__2,A)))) ) ) ) ) )).

fof(t11_rcomp_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
           => k1_rcomp_1(A,B) = k2_xboole_0(k2_rcomp_1(A,B),k2_tarski(A,B)) ) ) ) )).

fof(t12_rcomp_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
           => k2_rcomp_1(B,A) = k1_xboole_0 ) ) ) )).

fof(t13_rcomp_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ~ r1_xreal_0(B,A)
           => k1_rcomp_1(B,A) = k1_xboole_0 ) ) ) )).

fof(t14_rcomp_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k1_rcomp_1(A,A) = k1_seq_4(A) ) )).

fof(t15_rcomp_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ~ ( ~ r1_xreal_0(B,A)
                & k2_rcomp_1(A,B) = k1_xboole_0 )
            & ( r1_xreal_0(A,B)
             => ( r2_hidden(A,k1_rcomp_1(A,B))
                & r2_hidden(B,k1_rcomp_1(A,B)) ) )
            & r1_tarski(k2_rcomp_1(A,B),k1_rcomp_1(A,B)) ) ) ) )).

fof(t16_rcomp_1,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(B,C))
                      & r2_hidden(D,k1_rcomp_1(B,C)) )
                   => r1_tarski(k1_rcomp_1(A,D),k1_rcomp_1(B,C)) ) ) ) ) ) )).

fof(t17_rcomp_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( ( r2_hidden(A,k2_rcomp_1(B,C))
                      & r2_hidden(D,k2_rcomp_1(B,C)) )
                   => r1_tarski(k1_rcomp_1(A,D),k2_rcomp_1(B,C)) ) ) ) ) ) )).

fof(t18_rcomp_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
           => k1_rcomp_1(A,B) = k4_subset_1(k1_numbers,k1_rcomp_1(A,B),k1_rcomp_1(B,A)) ) ) ) )).

fof(d3_rcomp_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ( v1_rcomp_1(A)
      <=> ! [B] :
            ( ( v1_funct_1(B)
              & v1_funct_2(B,k5_numbers,k1_numbers)
              & m2_relset_1(B,k5_numbers,k1_numbers) )
           => ~ ( r1_tarski(k2_relat_1(B),A)
                & ! [C] :
                    ( ( v1_funct_1(C)
                      & v1_funct_2(C,k5_numbers,k1_numbers)
                      & m2_relset_1(C,k5_numbers,k1_numbers) )
                   => ~ ( m1_seqm_3(C,B)
                        & v4_seq_2(C)
                        & r2_hidden(k2_seq_2(C),A) ) ) ) ) ) ) )).

fof(d4_rcomp_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ( v2_rcomp_1(A)
      <=> ! [B] :
            ( ( v1_funct_1(B)
              & v1_funct_2(B,k5_numbers,k1_numbers)
              & m2_relset_1(B,k5_numbers,k1_numbers) )
           => ( ( r1_tarski(k2_relat_1(B),A)
                & v4_seq_2(B) )
             => r2_hidden(k2_seq_2(B),A) ) ) ) ) )).

fof(d5_rcomp_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ( v3_rcomp_1(A)
      <=> v2_rcomp_1(k3_subset_1(k1_numbers,A)) ) ) )).

fof(t19_rcomp_1,axiom,(
    $true )).

fof(t20_rcomp_1,axiom,(
    $true )).

fof(t21_rcomp_1,axiom,(
    $true )).

fof(t22_rcomp_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( r1_tarski(k2_relat_1(C),k1_rcomp_1(A,B))
               => v3_seq_2(C) ) ) ) ) )).

fof(t23_rcomp_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => v2_rcomp_1(k1_rcomp_1(A,B)) ) ) )).

fof(t24_rcomp_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => v1_rcomp_1(k1_rcomp_1(A,B)) ) ) )).

fof(t25_rcomp_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => v3_rcomp_1(k2_rcomp_1(A,B)) ) ) )).

fof(t26_rcomp_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ( v1_rcomp_1(A)
       => v2_rcomp_1(A) ) ) )).

fof(t27_rcomp_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
         => ( ! [C] :
                ( v1_xreal_0(C)
               => ~ ( r2_hidden(C,B)
                    & ! [D] :
                        ( v1_xreal_0(D)
                       => ! [E] :
                            ( m2_subset_1(E,k1_numbers,k5_numbers)
                           => ~ ( ~ r1_xreal_0(D,np__0)
                                & ! [F] :
                                    ( m2_subset_1(F,k1_numbers,k5_numbers)
                                   => ~ ( ~ r1_xreal_0(F,E)
                                        & r1_xreal_0(k18_complex1(k6_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,A,F),C)),D) ) ) ) ) ) ) )
           => ! [C] :
                ( ( v1_funct_1(C)
                  & v1_funct_2(C,k5_numbers,k1_numbers)
                  & m2_relset_1(C,k5_numbers,k1_numbers) )
               => ~ ( m1_seqm_3(C,A)
                    & v4_seq_2(C)
                    & r2_hidden(k2_seq_2(C),B) ) ) ) ) ) )).

fof(t28_rcomp_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ( v1_rcomp_1(A)
       => v3_seq_4(A) ) ) )).

fof(t29_rcomp_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ( ( v3_seq_4(A)
          & v2_rcomp_1(A) )
       => v1_rcomp_1(A) ) ) )).

fof(t30_rcomp_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ( ( v2_rcomp_1(A)
          & v1_seq_4(A) )
       => ( A = k1_xboole_0
          | r2_hidden(k4_seq_4(A),A) ) ) ) )).

fof(t31_rcomp_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ( ( v2_rcomp_1(A)
          & v2_seq_4(A) )
       => ( A = k1_xboole_0
          | r2_hidden(k5_seq_4(A),A) ) ) ) )).

fof(t32_rcomp_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ( v1_rcomp_1(A)
       => ( A = k1_xboole_0
          | ( r2_hidden(k4_seq_4(A),A)
            & r2_hidden(k5_seq_4(A),A) ) ) ) ) )).

fof(t33_rcomp_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ~ ( v1_rcomp_1(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) ) ) )
          & ! [B] :
              ( v1_xreal_0(B)
             => ! [C] :
                  ( v1_xreal_0(C)
                 => A != k1_rcomp_1(B,C) ) ) ) ) )).

fof(d6_rcomp_1,axiom,(
    $true )).

fof(d7_rcomp_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
         => ( m1_rcomp_1(B,A)
          <=> ? [C] :
                ( v1_xreal_0(C)
                & ~ r1_xreal_0(C,np__0)
                & B = k2_rcomp_1(k6_xcmplx_0(A,C),k2_xcmplx_0(A,C)) ) ) ) ) )).

fof(t34_rcomp_1,axiom,(
    $true )).

fof(t35_rcomp_1,axiom,(
    $true )).

fof(t36_rcomp_1,axiom,(
    $true )).

fof(t37_rcomp_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m1_rcomp_1(B,A)
         => r2_hidden(A,B) ) ) )).

fof(t38_rcomp_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m1_rcomp_1(B,A)
         => ! [C] :
              ( m1_rcomp_1(C,A)
             => ? [D] :
                  ( m1_rcomp_1(D,A)
                  & r1_tarski(D,B)
                  & r1_tarski(D,C) ) ) ) ) )).

fof(t39_rcomp_1,axiom,(
    ! [A] :
      ( ( v3_rcomp_1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( v1_xreal_0(B)
         => ~ ( r2_hidden(B,A)
              & ! [C] :
                  ( m1_rcomp_1(C,B)
                 => ~ r1_tarski(C,A) ) ) ) ) )).

fof(t40_rcomp_1,axiom,(
    ! [A] :
      ( ( v3_rcomp_1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( v1_xreal_0(B)
         => ~ ( r2_hidden(B,A)
              & ! [C] :
                  ( v1_xreal_0(C)
                 => ~ ( ~ r1_xreal_0(C,np__0)
                      & r1_tarski(k2_rcomp_1(k6_xcmplx_0(B,C),k2_xcmplx_0(B,C)),A) ) ) ) ) ) )).

fof(t41_rcomp_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ( ! [B] :
            ( v1_xreal_0(B)
           => ~ ( r2_hidden(B,A)
                & ! [C] :
                    ( m1_rcomp_1(C,B)
                   => ~ r1_tarski(C,A) ) ) )
       => v3_rcomp_1(A) ) ) )).

fof(t42_rcomp_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ( ! [B] :
            ( v1_xreal_0(B)
           => ~ ( r2_hidden(B,A)
                & ! [C] :
                    ( m1_rcomp_1(C,B)
                   => ~ r1_tarski(C,A) ) ) )
      <=> v3_rcomp_1(A) ) ) )).

fof(t43_rcomp_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ~ ( v3_rcomp_1(A)
          & v1_seq_4(A)
          & r2_hidden(k4_seq_4(A),A) ) ) )).

fof(t44_rcomp_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ~ ( v3_rcomp_1(A)
          & v2_seq_4(A)
          & r2_hidden(k5_seq_4(A),A) ) ) )).

fof(t45_rcomp_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ~ ( v3_rcomp_1(A)
          & v3_seq_4(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) ) ) )
          & ! [B] :
              ( v1_xreal_0(B)
             => ! [C] :
                  ( v1_xreal_0(C)
                 => A != k2_rcomp_1(B,C) ) ) ) ) )).

fof(t46_rcomp_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => r1_xboole_0(k2_rcomp_1(A,B),k2_tarski(A,B)) ) ) )).

fof(t47_rcomp_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( r2_hidden(C,k2_rcomp_1(A,B))
              <=> ( ~ r1_xreal_0(C,A)
                  & ~ r1_xreal_0(B,C) ) ) ) ) ) )).

fof(t48_rcomp_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( r2_hidden(C,k1_rcomp_1(A,B))
              <=> ( r1_xreal_0(A,C)
                  & r1_xreal_0(C,B) ) ) ) ) ) )).

fof(s1_rcomp_1,axiom,
    ( ! [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ? [B] :
            ( v1_xreal_0(B)
            & p1_s1_rcomp_1(A,B) ) )
   => ? [A] :
        ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers)
        & ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => p1_s1_rcomp_1(B,k2_seq_1(k5_numbers,k1_numbers,A,B)) ) ) )).

fof(dt_m1_rcomp_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m1_rcomp_1(B,A)
         => m1_subset_1(B,k1_zfmisc_1(k1_numbers)) ) ) )).

fof(existence_m1_rcomp_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ? [B] : m1_rcomp_1(B,A) ) )).

fof(dt_k1_rcomp_1,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => m1_subset_1(k1_rcomp_1(A,B),k1_zfmisc_1(k1_numbers)) ) )).

fof(dt_k2_rcomp_1,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => m1_subset_1(k2_rcomp_1(A,B),k1_zfmisc_1(k1_numbers)) ) )).

fof(d1_rcomp_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => k1_rcomp_1(A,B) = a_2_0_rcomp_1(A,B) ) ) )).

fof(d2_rcomp_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => k2_rcomp_1(A,B) = a_2_1_rcomp_1(A,B) ) ) )).

fof(fraenkel_a_2_0_rcomp_1,axiom,(
    ! [A,B,C] :
      ( ( v1_xreal_0(B)
        & v1_xreal_0(C) )
     => ( r2_hidden(A,a_2_0_rcomp_1(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,k1_numbers)
            & A = D
            & r1_xreal_0(B,D)
            & r1_xreal_0(D,C) ) ) ) )).

fof(fraenkel_a_2_1_rcomp_1,axiom,(
    ! [A,B,C] :
      ( ( v1_xreal_0(B)
        & v1_xreal_0(C) )
     => ( r2_hidden(A,a_2_1_rcomp_1(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,k1_numbers)
            & A = D
            & ~ r1_xreal_0(D,B)
            & ~ r1_xreal_0(C,D) ) ) ) )).
%------------------------------------------------------------------------------