SET007 Axioms: SET007+117.ax


%------------------------------------------------------------------------------
% File     : SET007+117 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Group and Field Definitions
% 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    : realset1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   48 (   2 unit)
%            Number of atoms       :  213 (  11 equality)
%            Maximal formula depth :   15 (   7 average)
%            Number of connectives :  212 (  47 ~  ;   0  |;  85  &)
%                                         (   9 <=>;  71 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   19 (   1 propositional; 0-3 arity)
%            Number of functors    :   13 (   1 constant; 0-3 arity)
%            Number of variables   :  126 (   0 singleton; 110 !;  16 ?)
%            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_realset1,axiom,(
    ! [A,B] :
      ( v1_relat_1(A)
     => v1_relat_1(k1_realset1(A,B)) ) )).

fof(fc2_realset1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v1_relat_1(k1_realset1(A,B))
        & v1_funct_1(k1_realset1(A,B)) ) ) )).

fof(rc1_realset1,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & v1_realset1(A) ) )).

fof(rc2_realset1,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & ~ v1_realset1(A) ) )).

fof(cc1_realset1,axiom,(
    ! [A] :
      ( ~ v1_realset1(A)
     => ~ v1_xboole_0(A) ) )).

fof(fc3_realset1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(k1_tarski(A))
      & v1_finset_1(k1_tarski(A))
      & v1_realset1(k1_tarski(A)) ) )).

fof(fc4_realset1,axiom,(
    ! [A,B] :
      ( ( ~ v1_realset1(A)
        & m4_realset1(B,A) )
     => ~ v1_xboole_0(k4_xboole_0(A,B)) ) )).

fof(rc3_realset1,axiom,(
    ! [A] :
      ( ~ v1_realset1(A)
     => ? [B] :
          ( m4_realset1(B,A)
          & ~ v1_xboole_0(B) ) ) )).

fof(t1_realset1,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(C)
        & v1_funct_2(C,k2_zfmisc_1(A,A),A)
        & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
     => ( r2_hidden(B,k2_zfmisc_1(A,A))
       => r2_hidden(k1_funct_1(C,B),A) ) ) )).

fof(t2_realset1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(A,A),A)
        & m2_relset_1(B,k2_zfmisc_1(A,A),A) )
     => ? [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
          & ! [D] :
              ( r2_hidden(D,k2_zfmisc_1(C,C))
             => r2_hidden(k1_funct_1(B,D),C) ) ) ) )).

fof(d1_realset1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(A,A),A)
        & m2_relset_1(B,k2_zfmisc_1(A,A),A) )
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ( r1_realset1(A,B,C)
          <=> ! [D] :
                ( r2_hidden(D,k2_zfmisc_1(C,C))
               => r2_hidden(k1_funct_1(B,D),C) ) ) ) ) )).

fof(d2_realset1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(A,A),A)
        & m2_relset_1(B,k2_zfmisc_1(A,A),A) )
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ( m1_realset1(C,A,B)
          <=> ! [D] :
                ( r2_hidden(D,k2_zfmisc_1(C,C))
               => r2_hidden(k1_funct_1(B,D),C) ) ) ) ) )).

fof(d3_realset1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] : k1_realset1(A,B) = k7_relat_1(A,k2_zfmisc_1(B,B)) ) )).

fof(t3_realset1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(A,A),A)
        & m2_relset_1(B,k2_zfmisc_1(A,A),A) )
     => ! [C] :
          ( m1_realset1(C,A,B)
         => ( v1_funct_1(k1_realset1(B,C))
            & v1_funct_2(k1_realset1(B,C),k2_zfmisc_1(C,C),C)
            & m2_relset_1(k1_realset1(B,C),k2_zfmisc_1(C,C),C) ) ) ) )).

fof(d4_realset1,axiom,(
    ! [A] :
      ( v1_realset1(A)
    <=> ~ ( ~ v1_xboole_0(A)
          & ! [B] : A != k1_tarski(B) ) ) )).

fof(t4_realset1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ~ v1_realset1(A)
      <=> ! [B] : ~ v1_xboole_0(k4_xboole_0(A,k1_tarski(B))) ) ) )).

fof(t5_realset1,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & ! [B] :
          ( m1_subset_1(B,A)
         => ~ v1_xboole_0(k4_xboole_0(A,k1_tarski(B))) ) ) )).

fof(t6_realset1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ~ ( ! [B] :
              ( m1_subset_1(B,A)
             => ~ v1_xboole_0(k4_xboole_0(A,k1_tarski(B))) )
          & v1_realset1(A) ) ) )).

fof(t7_realset1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ! [B] :
            ( m1_subset_1(B,A)
           => ~ v1_xboole_0(k4_xboole_0(A,k1_tarski(B))) )
       => ~ v1_realset1(A) ) ) )).

fof(d5_realset1,axiom,(
    ! [A] :
      ( ~ v1_realset1(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(A,A),A)
            & m2_relset_1(B,k2_zfmisc_1(A,A),A) )
         => ! [C] :
              ( m1_subset_1(C,A)
             => ( r2_realset1(A,B,C)
              <=> ( m1_realset1(k4_xboole_0(A,k1_tarski(C)),A,B)
                  & v1_funct_1(k4_xboole_0(k1_realset1(B,A),k1_tarski(C)))
                  & v1_funct_2(k4_xboole_0(k1_realset1(B,A),k1_tarski(C)),k2_zfmisc_1(k4_xboole_0(A,k1_tarski(C)),k4_xboole_0(A,k1_tarski(C))),k4_xboole_0(A,k1_tarski(C)))
                  & m2_relset_1(k4_xboole_0(k1_realset1(B,A),k1_tarski(C)),k2_zfmisc_1(k4_xboole_0(A,k1_tarski(C)),k4_xboole_0(A,k1_tarski(C))),k4_xboole_0(A,k1_tarski(C))) ) ) ) ) ) )).

fof(t8_realset1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ? [C] :
          ( v1_funct_1(C)
          & v1_funct_2(C,k2_zfmisc_1(A,A),A)
          & m2_relset_1(C,k2_zfmisc_1(A,A),A)
          & ! [D] :
              ( r2_hidden(D,k2_zfmisc_1(B,B))
             => r2_hidden(k1_funct_1(C,D),B) ) ) ) )).

fof(d6_realset1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k2_zfmisc_1(A,A),A)
            & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
         => ( m2_realset1(C,A,B)
          <=> ! [D] :
                ( r2_hidden(D,k2_zfmisc_1(B,B))
               => r2_hidden(k1_funct_1(C,D),B) ) ) ) ) )).

fof(t9_realset1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m2_realset1(C,A,B)
         => ( v1_funct_1(k1_realset1(C,B))
            & v1_funct_2(k1_realset1(C,B),k2_zfmisc_1(B,B),B)
            & m2_relset_1(k1_realset1(C,B),k2_zfmisc_1(B,B),B) ) ) ) )).

fof(d7_realset1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m2_realset1(C,A,B)
         => k3_realset1(A,B,C) = k1_realset1(C,B) ) ) )).

fof(t10_realset1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,A)
     => ? [C] :
          ( v1_funct_1(C)
          & v1_funct_2(C,k2_zfmisc_1(A,A),A)
          & m2_relset_1(C,k2_zfmisc_1(A,A),A)
          & ! [D] :
              ( r2_hidden(D,k2_zfmisc_1(k4_xboole_0(A,k1_tarski(B)),k4_xboole_0(A,k1_tarski(B))))
             => r2_hidden(k1_funct_1(C,D),k4_xboole_0(A,k1_tarski(B))) ) ) ) )).

fof(d8_realset1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,A)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k2_zfmisc_1(A,A),A)
            & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
         => ( m3_realset1(C,A,B)
          <=> ! [D] :
                ( r2_hidden(D,k2_zfmisc_1(k4_xboole_0(A,k1_tarski(B)),k4_xboole_0(A,k1_tarski(B))))
               => r2_hidden(k1_funct_1(C,D),k4_xboole_0(A,k1_tarski(B))) ) ) ) ) )).

fof(t11_realset1,axiom,(
    ! [A] :
      ( ~ v1_realset1(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m3_realset1(C,A,B)
             => ( v1_funct_1(k1_realset1(C,k4_xboole_0(A,k1_tarski(B))))
                & v1_funct_2(k1_realset1(C,k4_xboole_0(A,k1_tarski(B))),k2_zfmisc_1(k4_xboole_0(A,k1_tarski(B)),k4_xboole_0(A,k1_tarski(B))),k4_xboole_0(A,k1_tarski(B)))
                & m2_relset_1(k1_realset1(C,k4_xboole_0(A,k1_tarski(B))),k2_zfmisc_1(k4_xboole_0(A,k1_tarski(B)),k4_xboole_0(A,k1_tarski(B))),k4_xboole_0(A,k1_tarski(B))) ) ) ) ) )).

fof(d9_realset1,axiom,(
    ! [A] :
      ( ~ v1_realset1(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m3_realset1(C,A,B)
             => k4_realset1(A,B,C) = k1_realset1(C,k4_xboole_0(A,k1_tarski(B))) ) ) ) )).

fof(d10_realset1,axiom,(
    ! [A] :
      ( ~ v1_realset1(A)
     => ! [B] :
          ( m4_realset1(B,A)
        <=> ? [C] :
              ( m1_subset_1(C,A)
              & B = k1_tarski(C) ) ) ) )).

fof(t12_realset1,axiom,(
    ! [A] :
      ( ~ v1_realset1(A)
     => ! [B] :
          ( m4_realset1(B,A)
         => ~ v1_xboole_0(k4_xboole_0(A,B)) ) ) )).

fof(t13_realset1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ( v1_realset1(A)
      <=> ~ r1_xreal_0(np__2,k4_card_1(A)) ) ) )).

fof(t14_realset1,axiom,(
    ! [A] :
      ( ~ ( ~ v1_realset1(A)
          & ! [B,C] :
              ~ ( r2_hidden(B,A)
                & r2_hidden(C,A)
                & B != C ) )
      & ~ ( ? [B,C] :
              ( r2_hidden(B,A)
              & r2_hidden(C,A)
              & B != C )
          & v1_realset1(A) ) ) )).

fof(t15_realset1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ( ~ ( ~ v1_realset1(B)
            & ! [C] :
                ( m1_subset_1(C,A)
               => ! [D] :
                    ( m1_subset_1(D,A)
                   => ~ ( r2_hidden(C,B)
                        & r2_hidden(D,B)
                        & C != D ) ) ) )
        & ~ ( ? [C] :
                ( m1_subset_1(C,A)
                & ? [D] :
                    ( m1_subset_1(D,A)
                    & r2_hidden(C,B)
                    & r2_hidden(D,B)
                    & C != D ) )
            & v1_realset1(B) ) ) ) )).

fof(dt_m1_realset1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(A,A),A)
        & m1_relset_1(B,k2_zfmisc_1(A,A),A) )
     => ! [C] :
          ( m1_realset1(C,A,B)
         => m1_subset_1(C,k1_zfmisc_1(A)) ) ) )).

fof(existence_m1_realset1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(A,A),A)
        & m1_relset_1(B,k2_zfmisc_1(A,A),A) )
     => ? [C] : m1_realset1(C,A,B) ) )).

fof(dt_m2_realset1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m2_realset1(C,A,B)
         => ( v1_funct_1(C)
            & v1_funct_2(C,k2_zfmisc_1(A,A),A)
            & m2_relset_1(C,k2_zfmisc_1(A,A),A) ) ) ) )).

fof(existence_m2_realset1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ? [C] : m2_realset1(C,A,B) ) )).

fof(dt_m3_realset1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,A)
     => ! [C] :
          ( m3_realset1(C,A,B)
         => ( v1_funct_1(C)
            & v1_funct_2(C,k2_zfmisc_1(A,A),A)
            & m2_relset_1(C,k2_zfmisc_1(A,A),A) ) ) ) )).

fof(existence_m3_realset1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,A)
     => ? [C] : m3_realset1(C,A,B) ) )).

fof(dt_m4_realset1,axiom,(
    $true )).

fof(existence_m4_realset1,axiom,(
    ! [A] :
      ( ~ v1_realset1(A)
     => ? [B] : m4_realset1(B,A) ) )).

fof(dt_k1_realset1,axiom,(
    $true )).

fof(dt_k2_realset1,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(A,A),A)
        & m1_relset_1(B,k2_zfmisc_1(A,A),A)
        & m1_realset1(C,A,B) )
     => ( v1_funct_1(k2_realset1(A,B,C))
        & v1_funct_2(k2_realset1(A,B,C),k2_zfmisc_1(C,C),C)
        & m2_relset_1(k2_realset1(A,B,C),k2_zfmisc_1(C,C),C) ) ) )).

fof(redefinition_k2_realset1,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(A,A),A)
        & m1_relset_1(B,k2_zfmisc_1(A,A),A)
        & m1_realset1(C,A,B) )
     => k2_realset1(A,B,C) = k1_realset1(B,C) ) )).

fof(dt_k3_realset1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(B,k1_zfmisc_1(A))
        & m2_realset1(C,A,B) )
     => ( v1_funct_1(k3_realset1(A,B,C))
        & v1_funct_2(k3_realset1(A,B,C),k2_zfmisc_1(B,B),B)
        & m2_relset_1(k3_realset1(A,B,C),k2_zfmisc_1(B,B),B) ) ) )).

fof(dt_k4_realset1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_realset1(A)
        & m1_subset_1(B,A)
        & m3_realset1(C,A,B) )
     => ( v1_funct_1(k4_realset1(A,B,C))
        & v1_funct_2(k4_realset1(A,B,C),k2_zfmisc_1(k4_xboole_0(A,k1_tarski(B)),k4_xboole_0(A,k1_tarski(B))),k4_xboole_0(A,k1_tarski(B)))
        & m2_relset_1(k4_realset1(A,B,C),k2_zfmisc_1(k4_xboole_0(A,k1_tarski(B)),k4_xboole_0(A,k1_tarski(B))),k4_xboole_0(A,k1_tarski(B))) ) ) )).

fof(dt_k5_realset1,axiom,(
    ! [A,B] :
      ( ( ~ v1_realset1(A)
        & m1_subset_1(B,A) )
     => m4_realset1(k5_realset1(A,B),A) ) )).

fof(redefinition_k5_realset1,axiom,(
    ! [A,B] :
      ( ( ~ v1_realset1(A)
        & m1_subset_1(B,A) )
     => k5_realset1(A,B) = k1_tarski(B) ) )).
%------------------------------------------------------------------------------