SET007 Axioms: SET007+32.ax


%------------------------------------------------------------------------------
% File     : SET007+32 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Boolean Domains
% 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    : finsub_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   59 (  22 unit)
%            Number of atoms       :  181 (  13 equality)
%            Maximal formula depth :    9 (   5 average)
%            Number of connectives :  147 (  25 ~  ;   0  |;  69  &)
%                                         (   7 <=>;  46 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   11 (   1 propositional; 0-2 arity)
%            Number of functors    :   12 (   1 constant; 0-3 arity)
%            Number of variables   :   96 (   0 singleton;  95 !;   1 ?)
%            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(cc1_finsub_1,axiom,(
    ! [A] :
      ( v4_finsub_1(A)
     => ( v1_finsub_1(A)
        & v3_finsub_1(A) ) ) )).

fof(cc2_finsub_1,axiom,(
    ! [A] :
      ( ( v1_finsub_1(A)
        & v3_finsub_1(A) )
     => v4_finsub_1(A) ) )).

fof(rc1_finsub_1,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & v1_finsub_1(A)
      & v2_finsub_1(A)
      & v3_finsub_1(A)
      & v4_finsub_1(A) ) )).

fof(fc1_finsub_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(k1_zfmisc_1(A))
      & v1_finsub_1(k1_zfmisc_1(A))
      & v3_finsub_1(k1_zfmisc_1(A))
      & v4_finsub_1(k1_zfmisc_1(A)) ) )).

fof(fc2_finsub_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(k5_finsub_1(A))
      & v1_finsub_1(k5_finsub_1(A))
      & v3_finsub_1(k5_finsub_1(A))
      & v4_finsub_1(k5_finsub_1(A)) ) )).

fof(cc3_finsub_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k5_finsub_1(A))
     => v1_finset_1(B) ) )).

fof(d1_finsub_1,axiom,(
    ! [A] :
      ( v1_finsub_1(A)
    <=> ! [B,C] :
          ( ( r2_hidden(B,A)
            & r2_hidden(C,A) )
         => r2_hidden(k2_xboole_0(B,C),A) ) ) )).

fof(d2_finsub_1,axiom,(
    ! [A] :
      ( v2_finsub_1(A)
    <=> ! [B,C] :
          ( ( r2_hidden(B,A)
            & r2_hidden(C,A) )
         => r2_hidden(k3_xboole_0(B,C),A) ) ) )).

fof(d3_finsub_1,axiom,(
    ! [A] :
      ( v3_finsub_1(A)
    <=> ! [B,C] :
          ( ( r2_hidden(B,A)
            & r2_hidden(C,A) )
         => r2_hidden(k4_xboole_0(B,C),A) ) ) )).

fof(d4_finsub_1,axiom,(
    ! [A] :
      ( v4_finsub_1(A)
    <=> ( v1_finsub_1(A)
        & v3_finsub_1(A) ) ) )).

fof(t1_finsub_1,axiom,(
    $true )).

fof(t2_finsub_1,axiom,(
    $true )).

fof(t3_finsub_1,axiom,(
    $true )).

fof(t4_finsub_1,axiom,(
    $true )).

fof(t5_finsub_1,axiom,(
    $true )).

fof(t6_finsub_1,axiom,(
    $true )).

fof(t7_finsub_1,axiom,(
    $true )).

fof(t8_finsub_1,axiom,(
    $true )).

fof(t9_finsub_1,axiom,(
    $true )).

fof(t10_finsub_1,axiom,(
    ! [A] :
      ( v4_finsub_1(A)
    <=> ! [B,C] :
          ( ( r2_hidden(B,A)
            & r2_hidden(C,A) )
         => ( r2_hidden(k2_xboole_0(B,C),A)
            & r2_hidden(k4_xboole_0(B,C),A) ) ) ) )).

fof(t11_finsub_1,axiom,(
    $true )).

fof(t12_finsub_1,axiom,(
    $true )).

fof(t13_finsub_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(C)
        & v4_finsub_1(C) )
     => ( ( m1_subset_1(A,C)
          & m1_subset_1(B,C) )
       => m1_subset_1(k3_xboole_0(A,B),C) ) ) )).

fof(t14_finsub_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(C)
        & v4_finsub_1(C) )
     => ( ( m1_subset_1(A,C)
          & m1_subset_1(B,C) )
       => m1_subset_1(k5_xboole_0(A,B),C) ) ) )).

fof(t15_finsub_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ! [B] :
            ( m1_subset_1(B,A)
           => ! [C] :
                ( m1_subset_1(C,A)
               => ( r2_hidden(k5_xboole_0(B,C),A)
                  & r2_hidden(k4_xboole_0(B,C),A) ) ) )
       => v4_finsub_1(A) ) ) )).

fof(t16_finsub_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ! [B] :
            ( m1_subset_1(B,A)
           => ! [C] :
                ( m1_subset_1(C,A)
               => ( r2_hidden(k5_xboole_0(B,C),A)
                  & r2_hidden(k3_xboole_0(B,C),A) ) ) )
       => v4_finsub_1(A) ) ) )).

fof(t17_finsub_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ! [B] :
            ( m1_subset_1(B,A)
           => ! [C] :
                ( m1_subset_1(C,A)
               => ( r2_hidden(k5_xboole_0(B,C),A)
                  & r2_hidden(k2_xboole_0(B,C),A) ) ) )
       => v4_finsub_1(A) ) ) )).

fof(t18_finsub_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_finsub_1(A) )
     => r2_hidden(k1_xboole_0,A) ) )).

fof(t19_finsub_1,axiom,(
    $true )).

fof(t20_finsub_1,axiom,(
    ! [A] : v4_finsub_1(k1_zfmisc_1(A)) )).

fof(t21_finsub_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_finsub_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v4_finsub_1(B) )
         => ( ~ v1_xboole_0(k3_xboole_0(A,B))
            & v4_finsub_1(k3_xboole_0(A,B)) ) ) ) )).

fof(d5_finsub_1,axiom,(
    ! [A,B] :
      ( v4_finsub_1(B)
     => ( B = k5_finsub_1(A)
      <=> ! [C] :
            ( r2_hidden(C,B)
          <=> ( r1_tarski(C,A)
              & v1_finset_1(C) ) ) ) ) )).

fof(t22_finsub_1,axiom,(
    $true )).

fof(t23_finsub_1,axiom,(
    ! [A,B] :
      ( r1_tarski(A,B)
     => r1_tarski(k5_finsub_1(A),k5_finsub_1(B)) ) )).

fof(t24_finsub_1,axiom,(
    ! [A,B] : k5_finsub_1(k3_xboole_0(A,B)) = k3_xboole_0(k5_finsub_1(A),k5_finsub_1(B)) )).

fof(t25_finsub_1,axiom,(
    ! [A,B] : r1_tarski(k2_xboole_0(k5_finsub_1(A),k5_finsub_1(B)),k5_finsub_1(k2_xboole_0(A,B))) )).

fof(t26_finsub_1,axiom,(
    ! [A] : r1_tarski(k5_finsub_1(A),k1_zfmisc_1(A)) )).

fof(t27_finsub_1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => k5_finsub_1(A) = k1_zfmisc_1(A) ) )).

fof(t28_finsub_1,axiom,(
    k5_finsub_1(k1_xboole_0) = k1_tarski(k1_xboole_0) )).

fof(t29_finsub_1,axiom,(
    $true )).

fof(t30_finsub_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k5_finsub_1(A))
     => v1_finset_1(B) ) )).

fof(t31_finsub_1,axiom,(
    $true )).

fof(t32_finsub_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k5_finsub_1(A))
     => m1_subset_1(B,k1_zfmisc_1(A)) ) )).

fof(t33_finsub_1,axiom,(
    $true )).

fof(t34_finsub_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ( v1_finset_1(A)
       => m1_subset_1(B,k5_finsub_1(A)) ) ) )).

fof(dt_k1_finsub_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v4_finsub_1(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => m1_subset_1(k1_finsub_1(A,B,C),A) ) )).

fof(commutativity_k1_finsub_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v4_finsub_1(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k1_finsub_1(A,B,C) = k1_finsub_1(A,C,B) ) )).

fof(idempotence_k1_finsub_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v4_finsub_1(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k1_finsub_1(A,B,B) = B ) )).

fof(redefinition_k1_finsub_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v4_finsub_1(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k1_finsub_1(A,B,C) = k2_xboole_0(B,C) ) )).

fof(dt_k2_finsub_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v4_finsub_1(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => m1_subset_1(k2_finsub_1(A,B,C),A) ) )).

fof(redefinition_k2_finsub_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v4_finsub_1(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k2_finsub_1(A,B,C) = k4_xboole_0(B,C) ) )).

fof(dt_k3_finsub_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v4_finsub_1(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => m1_subset_1(k3_finsub_1(A,B,C),A) ) )).

fof(commutativity_k3_finsub_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v4_finsub_1(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k3_finsub_1(A,B,C) = k3_finsub_1(A,C,B) ) )).

fof(idempotence_k3_finsub_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v4_finsub_1(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k3_finsub_1(A,B,B) = B ) )).

fof(redefinition_k3_finsub_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v4_finsub_1(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k3_finsub_1(A,B,C) = k3_xboole_0(B,C) ) )).

fof(dt_k4_finsub_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v4_finsub_1(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => m1_subset_1(k4_finsub_1(A,B,C),A) ) )).

fof(commutativity_k4_finsub_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v4_finsub_1(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k4_finsub_1(A,B,C) = k4_finsub_1(A,C,B) ) )).

fof(redefinition_k4_finsub_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v4_finsub_1(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k4_finsub_1(A,B,C) = k5_xboole_0(B,C) ) )).

fof(dt_k5_finsub_1,axiom,(
    ! [A] : v4_finsub_1(k5_finsub_1(A)) )).
%------------------------------------------------------------------------------