SET007 Axioms: SET007+31.ax


%------------------------------------------------------------------------------
% File     : SET007+31 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Finite Sets
% 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    : finset_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   57 (  18 unit)
%            Number of atoms       :  169 (  10 equality)
%            Maximal formula depth :   13 (   5 average)
%            Number of connectives :  129 (  17 ~  ;   2  |;  63  &)
%                                         (   4 <=>;  43 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   17 (   1 propositional; 0-2 arity)
%            Number of functors    :   28 (   4 constant; 0-8 arity)
%            Number of variables   :  115 (   0 singleton; 109 !;   6 ?)
%            Maximal term depth    :    3 (   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(rc1_finset_1,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & v1_finset_1(A) ) )).

fof(cc1_finset_1,axiom,(
    ! [A] :
      ( v1_xboole_0(A)
     => v1_finset_1(A) ) )).

fof(rc2_finset_1,axiom,(
    ! [A] :
    ? [B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
      & v1_xboole_0(B)
      & v1_relat_1(B)
      & v1_funct_1(B)
      & v2_funct_1(B)
      & v1_ordinal1(B)
      & v2_ordinal1(B)
      & v3_ordinal1(B)
      & v4_ordinal2(B)
      & v1_finset_1(B) ) )).

fof(fc1_finset_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(k1_tarski(A))
      & v1_finset_1(k1_tarski(A)) ) )).

fof(rc3_finset_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
          & ~ v1_xboole_0(B)
          & v1_finset_1(B) ) ) )).

fof(fc2_finset_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(k2_tarski(A,B))
      & v1_finset_1(k2_tarski(A,B)) ) )).

fof(fc3_finset_1,axiom,(
    ! [A,B,C] : v1_finset_1(k1_enumset1(A,B,C)) )).

fof(fc4_finset_1,axiom,(
    ! [A,B,C,D] : v1_finset_1(k2_enumset1(A,B,C,D)) )).

fof(fc5_finset_1,axiom,(
    ! [A,B,C,D,E] : v1_finset_1(k3_enumset1(A,B,C,D,E)) )).

fof(fc6_finset_1,axiom,(
    ! [A,B,C,D,E,F] : v1_finset_1(k4_enumset1(A,B,C,D,E,F)) )).

fof(fc7_finset_1,axiom,(
    ! [A,B,C,D,E,F,G] : v1_finset_1(k5_enumset1(A,B,C,D,E,F,G)) )).

fof(fc8_finset_1,axiom,(
    ! [A,B,C,D,E,F,G,H] : v1_finset_1(k6_enumset1(A,B,C,D,E,F,G,H)) )).

fof(cc2_finset_1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => v1_finset_1(B) ) ) )).

fof(fc9_finset_1,axiom,(
    ! [A,B] :
      ( ( v1_finset_1(A)
        & v1_finset_1(B) )
     => v1_finset_1(k2_xboole_0(A,B)) ) )).

fof(fc10_finset_1,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => v1_finset_1(k3_xboole_0(A,B)) ) )).

fof(fc11_finset_1,axiom,(
    ! [A,B] :
      ( v1_finset_1(A)
     => v1_finset_1(k3_xboole_0(A,B)) ) )).

fof(fc12_finset_1,axiom,(
    ! [A,B] :
      ( v1_finset_1(A)
     => v1_finset_1(k4_xboole_0(A,B)) ) )).

fof(fc13_finset_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finset_1(B) )
     => v1_finset_1(k9_relat_1(A,B)) ) )).

fof(fc14_finset_1,axiom,(
    ! [A,B] :
      ( ( v1_finset_1(A)
        & v1_finset_1(B) )
     => v1_finset_1(k2_zfmisc_1(A,B)) ) )).

fof(fc15_finset_1,axiom,(
    ! [A,B,C] :
      ( ( v1_finset_1(A)
        & v1_finset_1(B)
        & v1_finset_1(C) )
     => v1_finset_1(k3_zfmisc_1(A,B,C)) ) )).

fof(fc16_finset_1,axiom,(
    ! [A,B,C,D] :
      ( ( v1_finset_1(A)
        & v1_finset_1(B)
        & v1_finset_1(C)
        & v1_finset_1(D) )
     => v1_finset_1(k4_zfmisc_1(A,B,C,D)) ) )).

fof(fc17_finset_1,axiom,(
    ! [A,B] :
      ( ( v1_finset_1(A)
        & v1_finset_1(B) )
     => v1_finset_1(k5_xboole_0(A,B)) ) )).

fof(rc4_finset_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
          & ~ v1_xboole_0(B)
          & v1_finset_1(B) ) ) )).

fof(d1_finset_1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
    <=> ? [B] :
          ( v1_relat_1(B)
          & v1_funct_1(B)
          & k2_relat_1(B) = A
          & r2_hidden(k1_relat_1(B),k5_ordinal2) ) ) )).

fof(t1_finset_1,axiom,(
    $true )).

fof(t2_finset_1,axiom,(
    $true )).

fof(t3_finset_1,axiom,(
    $true )).

fof(t4_finset_1,axiom,(
    $true )).

fof(t5_finset_1,axiom,(
    $true )).

fof(t6_finset_1,axiom,(
    $true )).

fof(t7_finset_1,axiom,(
    $true )).

fof(t8_finset_1,axiom,(
    $true )).

fof(t9_finset_1,axiom,(
    $true )).

fof(t10_finset_1,axiom,(
    $true )).

fof(t11_finset_1,axiom,(
    $true )).

fof(t12_finset_1,axiom,(
    $true )).

fof(t13_finset_1,axiom,(
    ! [A,B] :
      ( ( r1_tarski(A,B)
        & v1_finset_1(B) )
     => v1_finset_1(A) ) )).

fof(t14_finset_1,axiom,(
    ! [A,B] :
      ( ( v1_finset_1(A)
        & v1_finset_1(B) )
     => v1_finset_1(k2_xboole_0(A,B)) ) )).

fof(t15_finset_1,axiom,(
    ! [A,B] :
      ( v1_finset_1(A)
     => v1_finset_1(k3_xboole_0(A,B)) ) )).

fof(t16_finset_1,axiom,(
    ! [A,B] :
      ( v1_finset_1(A)
     => v1_finset_1(k4_xboole_0(A,B)) ) )).

fof(t17_finset_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( v1_finset_1(A)
       => v1_finset_1(k9_relat_1(B,A)) ) ) )).

fof(t18_finset_1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
         => ~ ( B != k1_xboole_0
              & ! [C] :
                  ~ ( r2_hidden(C,B)
                    & ! [D] :
                        ( ( r2_hidden(D,B)
                          & r1_tarski(C,D) )
                       => D = C ) ) ) ) ) )).

fof(t19_finset_1,axiom,(
    ! [A,B] :
      ( ( v1_finset_1(A)
        & v1_finset_1(B) )
     => v1_finset_1(k2_zfmisc_1(A,B)) ) )).

fof(t20_finset_1,axiom,(
    ! [A,B,C] :
      ( ( v1_finset_1(A)
        & v1_finset_1(B)
        & v1_finset_1(C) )
     => v1_finset_1(k3_zfmisc_1(A,B,C)) ) )).

fof(t21_finset_1,axiom,(
    ! [A,B,C,D] :
      ( ( v1_finset_1(A)
        & v1_finset_1(B)
        & v1_finset_1(C)
        & v1_finset_1(D) )
     => v1_finset_1(k4_zfmisc_1(A,B,C,D)) ) )).

fof(t22_finset_1,axiom,(
    ! [A,B] :
      ( v1_finset_1(k2_zfmisc_1(B,A))
     => ( A = k1_xboole_0
        | v1_finset_1(B) ) ) )).

fof(t23_finset_1,axiom,(
    ! [A,B] :
      ( v1_finset_1(k2_zfmisc_1(A,B))
     => ( A = k1_xboole_0
        | v1_finset_1(B) ) ) )).

fof(t24_finset_1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
    <=> v1_finset_1(k1_zfmisc_1(A)) ) )).

fof(t25_finset_1,axiom,(
    ! [A] :
      ( ( v1_finset_1(A)
        & ! [B] :
            ( r2_hidden(B,A)
           => v1_finset_1(B) ) )
    <=> v1_finset_1(k3_tarski(A)) ) )).

fof(t26_finset_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v1_finset_1(k1_relat_1(A))
       => v1_finset_1(k2_relat_1(A)) ) ) )).

fof(t27_finset_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( ( r1_tarski(A,k2_relat_1(B))
          & v1_finset_1(k10_relat_1(B,A)) )
       => v1_finset_1(A) ) ) )).

fof(t28_finset_1,axiom,(
    ! [A,B] :
      ( ( v1_finset_1(A)
        & v1_finset_1(B) )
     => v1_finset_1(k5_xboole_0(A,B)) ) )).

fof(t29_finset_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v1_finset_1(k1_relat_1(A))
      <=> v1_finset_1(A) ) ) )).

fof(t30_finset_1,axiom,(
    ! [A] :
      ~ ( v1_finset_1(A)
        & A != k1_xboole_0
        & v6_ordinal1(A)
        & ! [B] :
            ~ ( r2_hidden(B,A)
              & ! [C] :
                  ( r2_hidden(C,A)
                 => r1_tarski(B,C) ) ) ) )).

fof(t31_finset_1,axiom,(
    ! [A] :
      ~ ( v1_finset_1(A)
        & A != k1_xboole_0
        & v6_ordinal1(A)
        & ! [B] :
            ~ ( r2_hidden(B,A)
              & ! [C] :
                  ( r2_hidden(C,A)
                 => r1_tarski(C,B) ) ) ) )).

fof(s1_finset_1,axiom,(
    ? [A] :
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & k1_relat_1(A) = f1_s1_finset_1
      & ! [B] :
          ( v3_ordinal1(B)
         => ( r2_hidden(B,f1_s1_finset_1)
           => ( ( p1_s1_finset_1(B)
               => k1_funct_1(A,B) = f2_s1_finset_1(B) )
              & ( ~ p1_s1_finset_1(B)
               => k1_funct_1(A,B) = f3_s1_finset_1(B) ) ) ) ) ) )).

fof(s2_finset_1,axiom,
    ( ( v1_finset_1(f1_s2_finset_1)
      & p1_s2_finset_1(k1_xboole_0)
      & ! [A,B] :
          ( ( r2_hidden(A,f1_s2_finset_1)
            & r1_tarski(B,f1_s2_finset_1)
            & p1_s2_finset_1(B) )
         => p1_s2_finset_1(k2_xboole_0(B,k1_tarski(A))) ) )
   => p1_s2_finset_1(f1_s2_finset_1) )).
%------------------------------------------------------------------------------