SET007 Axioms: SET007+577.ax


%------------------------------------------------------------------------------
% File     : SET007+577 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Classes of Independent Partitions
% 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    : partit_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   25 (   2 unit)
%            Number of atoms       :  142 (   9 equality)
%            Maximal formula depth :   18 (   9 average)
%            Number of connectives :  137 (  20 ~  ;   0  |;  31  &)
%                                         (   5 <=>;  81 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   19 (   1 propositional; 0-4 arity)
%            Number of functors    :   18 (   1 constant; 0-6 arity)
%            Number of variables   :   84 (   0 singleton;  82 !;   2 ?)
%            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(d1_partit_2,axiom,(
    ! [A,B,C] :
      ( m2_relset_1(C,A,B)
     => ! [D] :
          ( m2_relset_1(D,A,B)
         => ( r1_partit_2(A,B,C,D)
          <=> ! [E] :
                ( m1_subset_1(E,A)
               => ! [F] :
                    ( m1_subset_1(F,B)
                   => ( r2_hidden(k4_tarski(E,F),C)
                     => r2_hidden(k4_tarski(E,F),D) ) ) ) ) ) ) )).

fof(t1_partit_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => k2_bvfunc_2(A,k1_subset_1(k1_bvfunc_2(A))) = k6_partit1(A) ) )).

fof(t2_partit_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v3_relat_2(B)
            & v8_relat_2(B)
            & v1_partfun1(B,A,A)
            & m2_relset_1(B,A,A) )
         => ! [C] :
              ( ( v3_relat_2(C)
                & v8_relat_2(C)
                & v1_partfun1(C,A,A)
                & m2_relset_1(C,A,A) )
             => r1_partit_2(A,A,k3_eqrel_1(A,B,C),k7_relset_1(A,A,A,A,B,C)) ) ) ) )).

fof(t3_partit_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_relset_1(B,A,A)
         => r1_partit_2(A,A,B,k1_eqrel_1(A)) ) ) )).

fof(t4_partit_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v3_relat_2(B)
            & v8_relat_2(B)
            & v1_partfun1(B,A,A)
            & m2_relset_1(B,A,A) )
         => ( k7_relset_1(A,A,A,A,k1_eqrel_1(A),B) = k1_eqrel_1(A)
            & k7_relset_1(A,A,A,A,B,k1_eqrel_1(A)) = k1_eqrel_1(A) ) ) ) )).

fof(t5_partit_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( m1_subset_1(D,A)
                 => ( r2_hidden(k4_tarski(C,D),k4_partit1(A,B))
                  <=> r2_hidden(C,k22_bvfunc_1(A,D,B)) ) ) ) ) ) )).

fof(t6_partit_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => ! [D] :
                  ( m1_eqrel_1(D,A)
                 => ( k4_partit1(A,D) = k7_relset_1(A,A,A,A,k4_partit1(A,B),k4_partit1(A,C))
                   => ! [E] :
                        ( m1_subset_1(E,A)
                       => ! [F] :
                            ( m1_subset_1(F,A)
                           => ( r2_hidden(E,k22_bvfunc_1(A,F,D))
                            <=> ? [G] :
                                  ( m1_subset_1(G,A)
                                  & r2_hidden(E,k22_bvfunc_1(A,G,B))
                                  & r2_hidden(G,k22_bvfunc_1(A,F,C)) ) ) ) ) ) ) ) ) ) )).

fof(t7_partit_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( v1_relat_1(B)
         => ! [C] :
              ( ( r1_relat_2(A,C)
                & r1_relat_2(B,C) )
             => r1_relat_2(k5_relat_1(A,B),C) ) ) ) )).

fof(t8_partit_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( r1_relat_2(A,B)
         => r1_tarski(B,k3_relat_1(A)) ) ) )).

fof(t9_partit_2,axiom,(
    ! [A,B] :
      ( m2_relset_1(B,A,A)
     => ( r1_relat_2(B,A)
       => A = k3_relat_1(B) ) ) )).

fof(t10_partit_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v3_relat_2(B)
            & v8_relat_2(B)
            & v1_partfun1(B,A,A)
            & m2_relset_1(B,A,A) )
         => ! [C] :
              ( ( v3_relat_2(C)
                & v8_relat_2(C)
                & v1_partfun1(C,A,A)
                & m2_relset_1(C,A,A) )
             => ( k7_relset_1(A,A,A,A,B,C) = k7_relset_1(A,A,A,A,C,B)
               => ( v3_relat_2(k7_relset_1(A,A,A,A,B,C))
                  & v8_relat_2(k7_relset_1(A,A,A,A,B,C))
                  & v1_partfun1(k7_relset_1(A,A,A,A,B,C),A,A)
                  & m2_relset_1(k7_relset_1(A,A,A,A,B,C),A,A) ) ) ) ) ) )).

fof(t11_partit_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
         => ! [C] :
              ( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
             => ( r1_bvfunc_1(A,B,C)
               => r1_bvfunc_1(A,k5_valuat_1(A,C),k5_valuat_1(A,B)) ) ) ) ) )).

fof(t12_partit_2,axiom,(
    $true )).

fof(t13_partit_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
         => ! [C] :
              ( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k1_bvfunc_2(A)))
                 => ! [E] :
                      ( m1_eqrel_1(E,A)
                     => ( r1_bvfunc_1(A,B,C)
                       => r1_bvfunc_1(A,k6_bvfunc_2(A,B,D,E),k6_bvfunc_2(A,C,D,E)) ) ) ) ) ) ) )).

fof(t14_partit_2,axiom,(
    $true )).

fof(t15_partit_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
         => ! [C] :
              ( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k1_bvfunc_2(A)))
                 => ! [E] :
                      ( m1_eqrel_1(E,A)
                     => ( r1_bvfunc_1(A,B,C)
                       => r1_bvfunc_1(A,k7_bvfunc_2(A,B,D,E),k7_bvfunc_2(A,C,D,E)) ) ) ) ) ) ) )).

fof(t16_partit_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A)))
         => ( v2_bvfunc_2(B,A)
           => ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
               => ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(k1_bvfunc_2(A)))
                   => ( ( r1_tarski(C,B)
                        & r1_tarski(D,B) )
                     => k7_relset_1(A,A,A,A,k4_partit1(A,k2_bvfunc_2(A,C)),k4_partit1(A,k2_bvfunc_2(A,D))) = k7_relset_1(A,A,A,A,k4_partit1(A,k2_bvfunc_2(A,D)),k4_partit1(A,k2_bvfunc_2(A,C))) ) ) ) ) ) ) )).

fof(t17_partit_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
             => ! [D] :
                  ( m1_eqrel_1(D,A)
                 => ! [E] :
                      ( m1_eqrel_1(E,A)
                     => ( v2_bvfunc_2(C,A)
                       => k6_bvfunc_2(A,k6_bvfunc_2(A,B,C,D),C,E) = k6_bvfunc_2(A,k6_bvfunc_2(A,B,C,E),C,D) ) ) ) ) ) ) )).

fof(t18_partit_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
             => ! [D] :
                  ( m1_eqrel_1(D,A)
                 => ! [E] :
                      ( m1_eqrel_1(E,A)
                     => ( v2_bvfunc_2(C,A)
                       => k7_bvfunc_2(A,k7_bvfunc_2(A,B,C,D),C,E) = k7_bvfunc_2(A,k7_bvfunc_2(A,B,C,E),C,D) ) ) ) ) ) ) )).

fof(t19_partit_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
             => ! [D] :
                  ( m1_eqrel_1(D,A)
                 => ! [E] :
                      ( m1_eqrel_1(E,A)
                     => ( v2_bvfunc_2(C,A)
                       => r1_bvfunc_1(A,k7_bvfunc_2(A,k6_bvfunc_2(A,B,C,D),C,E),k6_bvfunc_2(A,k7_bvfunc_2(A,B,C,E),C,D)) ) ) ) ) ) ) )).

fof(dt_m1_partit_2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A))) )
     => ! [C] :
          ( m1_partit_2(C,A,B)
         => m1_eqrel_1(C,A) ) ) )).

fof(existence_m1_partit_2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A))) )
     => ? [C] : m1_partit_2(C,A,B) ) )).

fof(redefinition_m1_partit_2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A))) )
     => ! [C] :
          ( m1_partit_2(C,A,B)
        <=> m1_subset_1(C,B) ) ) )).

fof(reflexivity_r1_partit_2,axiom,(
    ! [A,B,C,D] :
      ( ( m1_relset_1(C,A,B)
        & m1_relset_1(D,A,B) )
     => r1_partit_2(A,B,C,C) ) )).

fof(redefinition_r1_partit_2,axiom,(
    ! [A,B,C,D] :
      ( ( m1_relset_1(C,A,B)
        & m1_relset_1(D,A,B) )
     => ( r1_partit_2(A,B,C,D)
      <=> r1_tarski(C,D) ) ) )).
%------------------------------------------------------------------------------