SET007 Axioms: SET007+592.ax


%------------------------------------------------------------------------------
% File     : SET007+592 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Predicate Calculus for Boolean Valued Functions. Part II
% 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    : bvfunc_4 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   28 (   0 unit)
%            Number of atoms       :  168 (  34 equality)
%            Maximal formula depth :   15 (  10 average)
%            Number of connectives :  168 (  28 ~  ;   0  |;   9  &)
%                                         (   1 <=>; 130 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    9 (   0 propositional; 1-4 arity)
%            Number of functors    :   14 (   1 constant; 0-4 arity)
%            Number of variables   :  115 (   0 singleton; 115 !;   0 ?)
%            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(t1_bvfunc_4,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] :
                  ( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
                 => ( r1_bvfunc_1(A,B,k14_bvfunc_1(A,C,D))
                   => r1_bvfunc_1(A,k6_valuat_1(A,B,C),D) ) ) ) ) ) )).

fof(t2_bvfunc_4,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] :
                  ( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
                 => ( r1_bvfunc_1(A,k6_valuat_1(A,B,C),D)
                   => r1_bvfunc_1(A,B,k14_bvfunc_1(A,C,D)) ) ) ) ) ) )).

fof(t3_bvfunc_4,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))
             => k8_bvfunc_1(A,B,k6_valuat_1(A,B,C)) = B ) ) ) )).

fof(t4_bvfunc_4,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))
             => k6_valuat_1(A,B,k8_bvfunc_1(A,B,C)) = B ) ) ) )).

fof(t5_bvfunc_4,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
         => k6_valuat_1(A,B,k5_valuat_1(A,B)) = k18_bvfunc_1(A) ) ) )).

fof(t6_bvfunc_4,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
         => k8_bvfunc_1(A,B,k5_valuat_1(A,B)) = k19_bvfunc_1(A) ) ) )).

fof(t7_bvfunc_4,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))
             => k15_bvfunc_1(A,B,C) = k6_valuat_1(A,k14_bvfunc_1(A,B,C),k14_bvfunc_1(A,C,B)) ) ) ) )).

fof(t8_bvfunc_4,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))
             => k14_bvfunc_1(A,B,C) = k8_bvfunc_1(A,k5_valuat_1(A,B),C) ) ) ) )).

fof(t9_bvfunc_4,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))
             => k9_bvfunc_1(A,B,C) = k8_bvfunc_1(A,k6_valuat_1(A,k5_valuat_1(A,B),C),k6_valuat_1(A,B,k5_valuat_1(A,C))) ) ) ) )).

fof(t10_bvfunc_4,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))
             => ( k15_bvfunc_1(A,B,C) = k19_bvfunc_1(A)
              <=> ( k14_bvfunc_1(A,B,C) = k19_bvfunc_1(A)
                  & k14_bvfunc_1(A,C,B) = k19_bvfunc_1(A) ) ) ) ) ) )).

fof(t11_bvfunc_4,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] :
                  ( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
                 => ( ( k15_bvfunc_1(A,B,C) = k19_bvfunc_1(A)
                      & k15_bvfunc_1(A,C,D) = k19_bvfunc_1(A) )
                   => k15_bvfunc_1(A,B,D) = k19_bvfunc_1(A) ) ) ) ) ) )).

fof(t12_bvfunc_4,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))
             => ( k15_bvfunc_1(A,B,C) = k19_bvfunc_1(A)
               => k15_bvfunc_1(A,k5_valuat_1(A,B),k5_valuat_1(A,C)) = k19_bvfunc_1(A) ) ) ) ) )).

fof(t13_bvfunc_4,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] :
                  ( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
                 => ! [E] :
                      ( m2_fraenkel(E,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
                     => ( ( k15_bvfunc_1(A,B,C) = k19_bvfunc_1(A)
                          & k15_bvfunc_1(A,D,E) = k19_bvfunc_1(A) )
                       => k15_bvfunc_1(A,k6_valuat_1(A,B,D),k6_valuat_1(A,C,E)) = k19_bvfunc_1(A) ) ) ) ) ) ) )).

fof(t14_bvfunc_4,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] :
                  ( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
                 => ! [E] :
                      ( m2_fraenkel(E,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
                     => ( ( k15_bvfunc_1(A,B,C) = k19_bvfunc_1(A)
                          & k15_bvfunc_1(A,D,E) = k19_bvfunc_1(A) )
                       => k15_bvfunc_1(A,k14_bvfunc_1(A,B,D),k14_bvfunc_1(A,C,E)) = k19_bvfunc_1(A) ) ) ) ) ) ) )).

fof(t15_bvfunc_4,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] :
                  ( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
                 => ! [E] :
                      ( m2_fraenkel(E,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
                     => ( ( k15_bvfunc_1(A,B,C) = k19_bvfunc_1(A)
                          & k15_bvfunc_1(A,D,E) = k19_bvfunc_1(A) )
                       => k15_bvfunc_1(A,k8_bvfunc_1(A,B,D),k8_bvfunc_1(A,C,E)) = k19_bvfunc_1(A) ) ) ) ) ) ) )).

fof(t16_bvfunc_4,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] :
                  ( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
                 => ! [E] :
                      ( m2_fraenkel(E,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
                     => ( ( k15_bvfunc_1(A,B,C) = k19_bvfunc_1(A)
                          & k15_bvfunc_1(A,D,E) = k19_bvfunc_1(A) )
                       => k15_bvfunc_1(A,k15_bvfunc_1(A,B,D),k15_bvfunc_1(A,C,E)) = k19_bvfunc_1(A) ) ) ) ) ) ) )).

fof(t17_bvfunc_4,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)
                     => k6_bvfunc_2(A,k15_bvfunc_1(A,B,C),D,E) = k6_valuat_1(A,k6_bvfunc_2(A,k14_bvfunc_1(A,B,C),D,E),k6_bvfunc_2(A,k14_bvfunc_1(A,C,B),D,E)) ) ) ) ) ) )).

fof(t18_bvfunc_4,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)
                     => r1_bvfunc_1(A,k6_bvfunc_2(A,B,C,D),k7_bvfunc_2(A,B,C,E)) ) ) ) ) ) )).

fof(t19_bvfunc_4,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)
                     => ( ( v2_bvfunc_2(D,A)
                          & r2_hidden(E,D)
                          & r3_bvfunc_2(A,C,D,E)
                          & k14_bvfunc_1(A,B,C) = k19_bvfunc_1(A) )
                       => k14_bvfunc_1(A,k6_bvfunc_2(A,B,D,E),C) = k19_bvfunc_1(A) ) ) ) ) ) ) )).

fof(t20_bvfunc_4,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)
                 => ( r3_bvfunc_2(A,B,C,D)
                   => r1_bvfunc_1(A,k7_bvfunc_2(A,B,C,D),B) ) ) ) ) ) )).

fof(t21_bvfunc_4,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)
                 => ( r3_bvfunc_2(A,B,C,D)
                   => r1_bvfunc_1(A,B,k6_bvfunc_2(A,B,C,D)) ) ) ) ) ) )).

fof(t22_bvfunc_4,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)
                     => ( r3_bvfunc_2(A,B,C,E)
                       => r1_bvfunc_1(A,k6_bvfunc_2(A,B,C,D),k6_bvfunc_2(A,B,C,E)) ) ) ) ) ) ) )).

fof(t23_bvfunc_4,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)
                     => ( r3_bvfunc_2(A,B,C,D)
                       => r1_bvfunc_1(A,k7_bvfunc_2(A,B,C,D),k7_bvfunc_2(A,B,C,E)) ) ) ) ) ) ) )).

fof(t24_bvfunc_4,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,k6_bvfunc_2(A,k15_bvfunc_1(A,B,C),D,E),k15_bvfunc_1(A,k6_bvfunc_2(A,B,D,E),k6_bvfunc_2(A,C,D,E))) ) ) ) ) ) )).

fof(t25_bvfunc_4,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,k6_bvfunc_2(A,k6_valuat_1(A,B,C),D,E),k6_valuat_1(A,B,k6_bvfunc_2(A,C,D,E))) ) ) ) ) ) )).

fof(t26_bvfunc_4,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,k14_bvfunc_1(A,k6_bvfunc_2(A,B,D,E),C),k7_bvfunc_2(A,k14_bvfunc_1(A,B,C),D,E)) ) ) ) ) ) )).

fof(t27_bvfunc_4,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)
                     => ( k15_bvfunc_1(A,B,C) = k19_bvfunc_1(A)
                       => k15_bvfunc_1(A,k6_bvfunc_2(A,B,D,E),k6_bvfunc_2(A,C,D,E)) = k19_bvfunc_1(A) ) ) ) ) ) ) )).

fof(t28_bvfunc_4,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)
                     => ( k15_bvfunc_1(A,B,C) = k19_bvfunc_1(A)
                       => k15_bvfunc_1(A,k7_bvfunc_2(A,B,D,E),k7_bvfunc_2(A,C,D,E)) = k19_bvfunc_1(A) ) ) ) ) ) ) )).
%------------------------------------------------------------------------------