SET007 Axioms: SET007+633.ax


%------------------------------------------------------------------------------
% File     : SET007+633 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : 6 Variable Predicate Calculus for Boolean Valued Functions. Part I
% 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    : bvfunc23 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   11 (   0 unit)
%            Number of atoms       :  263 ( 176 equality)
%            Maximal formula depth :   42 (  33 average)
%            Number of connectives :  292 (  40 ~  ; 120  |;  50  &)
%                                         (   0 <=>;  82 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    8 (   0 propositional; 1-2 arity)
%            Number of functors    :   11 (   0 constant; 1-6 arity)
%            Number of variables   :  109 (   0 singleton; 109 !;   0 ?)
%            Maximal term depth    :    7 (   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_bvfunc23,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A)))
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => ! [D] :
                  ( m1_eqrel_1(D,A)
                 => ! [E] :
                      ( m1_eqrel_1(E,A)
                     => ! [F] :
                          ( m1_eqrel_1(F,A)
                         => ! [G] :
                              ( m1_eqrel_1(G,A)
                             => ! [H] :
                                  ( m1_eqrel_1(H,A)
                                 => ( B = k4_enumset1(C,D,E,F,G,H)
                                   => ( C = D
                                      | C = E
                                      | C = F
                                      | C = G
                                      | C = H
                                      | D = E
                                      | D = F
                                      | D = G
                                      | D = H
                                      | E = F
                                      | E = G
                                      | E = H
                                      | F = G
                                      | F = H
                                      | G = H
                                      | k5_bvfunc_2(A,C,B) = k2_partit1(A,k2_partit1(A,k2_partit1(A,k2_partit1(A,D,E),F),G),H) ) ) ) ) ) ) ) ) ) ) )).

fof(t2_bvfunc23,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A)))
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => ! [D] :
                  ( m1_eqrel_1(D,A)
                 => ! [E] :
                      ( m1_eqrel_1(E,A)
                     => ! [F] :
                          ( m1_eqrel_1(F,A)
                         => ! [G] :
                              ( m1_eqrel_1(G,A)
                             => ! [H] :
                                  ( m1_eqrel_1(H,A)
                                 => ( ( v2_bvfunc_2(B,A)
                                      & B = k4_enumset1(C,D,E,F,G,H) )
                                   => ( C = D
                                      | C = E
                                      | C = F
                                      | C = G
                                      | C = H
                                      | D = E
                                      | D = F
                                      | D = G
                                      | D = H
                                      | E = F
                                      | E = G
                                      | E = H
                                      | F = G
                                      | F = H
                                      | G = H
                                      | k5_bvfunc_2(A,D,B) = k2_partit1(A,k2_partit1(A,k2_partit1(A,k2_partit1(A,C,E),F),G),H) ) ) ) ) ) ) ) ) ) ) )).

fof(t3_bvfunc23,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A)))
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => ! [D] :
                  ( m1_eqrel_1(D,A)
                 => ! [E] :
                      ( m1_eqrel_1(E,A)
                     => ! [F] :
                          ( m1_eqrel_1(F,A)
                         => ! [G] :
                              ( m1_eqrel_1(G,A)
                             => ! [H] :
                                  ( m1_eqrel_1(H,A)
                                 => ( ( v2_bvfunc_2(B,A)
                                      & B = k4_enumset1(C,D,E,F,G,H) )
                                   => ( C = D
                                      | C = E
                                      | C = F
                                      | C = G
                                      | C = H
                                      | D = E
                                      | D = F
                                      | D = G
                                      | D = H
                                      | E = F
                                      | E = G
                                      | E = H
                                      | F = G
                                      | F = H
                                      | G = H
                                      | k5_bvfunc_2(A,E,B) = k2_partit1(A,k2_partit1(A,k2_partit1(A,k2_partit1(A,C,D),F),G),H) ) ) ) ) ) ) ) ) ) ) )).

fof(t4_bvfunc23,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A)))
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => ! [D] :
                  ( m1_eqrel_1(D,A)
                 => ! [E] :
                      ( m1_eqrel_1(E,A)
                     => ! [F] :
                          ( m1_eqrel_1(F,A)
                         => ! [G] :
                              ( m1_eqrel_1(G,A)
                             => ! [H] :
                                  ( m1_eqrel_1(H,A)
                                 => ( ( v2_bvfunc_2(B,A)
                                      & B = k4_enumset1(C,D,E,F,G,H) )
                                   => ( C = D
                                      | C = E
                                      | C = F
                                      | C = G
                                      | C = H
                                      | D = E
                                      | D = F
                                      | D = G
                                      | D = H
                                      | E = F
                                      | E = G
                                      | E = H
                                      | F = G
                                      | F = H
                                      | G = H
                                      | k5_bvfunc_2(A,F,B) = k2_partit1(A,k2_partit1(A,k2_partit1(A,k2_partit1(A,C,D),E),G),H) ) ) ) ) ) ) ) ) ) ) )).

fof(t5_bvfunc23,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A)))
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => ! [D] :
                  ( m1_eqrel_1(D,A)
                 => ! [E] :
                      ( m1_eqrel_1(E,A)
                     => ! [F] :
                          ( m1_eqrel_1(F,A)
                         => ! [G] :
                              ( m1_eqrel_1(G,A)
                             => ! [H] :
                                  ( m1_eqrel_1(H,A)
                                 => ( ( v2_bvfunc_2(B,A)
                                      & B = k4_enumset1(C,D,E,F,G,H) )
                                   => ( C = D
                                      | C = E
                                      | C = F
                                      | C = G
                                      | C = H
                                      | D = E
                                      | D = F
                                      | D = G
                                      | D = H
                                      | E = F
                                      | E = G
                                      | E = H
                                      | F = G
                                      | F = H
                                      | G = H
                                      | k5_bvfunc_2(A,G,B) = k2_partit1(A,k2_partit1(A,k2_partit1(A,k2_partit1(A,C,D),E),F),H) ) ) ) ) ) ) ) ) ) ) )).

fof(t6_bvfunc23,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A)))
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => ! [D] :
                  ( m1_eqrel_1(D,A)
                 => ! [E] :
                      ( m1_eqrel_1(E,A)
                     => ! [F] :
                          ( m1_eqrel_1(F,A)
                         => ! [G] :
                              ( m1_eqrel_1(G,A)
                             => ! [H] :
                                  ( m1_eqrel_1(H,A)
                                 => ( ( v2_bvfunc_2(B,A)
                                      & B = k4_enumset1(C,D,E,F,G,H) )
                                   => ( C = D
                                      | C = E
                                      | C = F
                                      | C = G
                                      | C = H
                                      | D = E
                                      | D = F
                                      | D = G
                                      | D = H
                                      | E = F
                                      | E = G
                                      | E = H
                                      | F = G
                                      | F = H
                                      | G = H
                                      | k5_bvfunc_2(A,H,B) = k2_partit1(A,k2_partit1(A,k2_partit1(A,k2_partit1(A,C,D),E),F),G) ) ) ) ) ) ) ) ) ) ) )).

fof(t7_bvfunc23,axiom,(
    ! [A,B,C,D,E,F,G] :
      ( ( v1_relat_1(G)
        & v1_funct_1(G) )
     => ! [H,I,J,K,L,M] :
          ( G = k1_funct_4(k1_funct_4(k1_funct_4(k1_funct_4(k1_funct_4(k3_cqc_lang(B,I),k3_cqc_lang(C,J)),k3_cqc_lang(D,K)),k3_cqc_lang(E,L)),k3_cqc_lang(F,M)),k3_cqc_lang(A,H))
         => ( A = B
            | A = C
            | A = D
            | A = E
            | A = F
            | B = C
            | B = D
            | B = E
            | B = F
            | C = D
            | C = E
            | C = F
            | D = E
            | D = F
            | E = F
            | ( k1_funct_1(G,A) = H
              & k1_funct_1(G,B) = I
              & k1_funct_1(G,C) = J
              & k1_funct_1(G,D) = K
              & k1_funct_1(G,E) = L
              & k1_funct_1(G,F) = M ) ) ) ) )).

fof(t8_bvfunc23,axiom,(
    ! [A,B,C,D,E,F,G] :
      ( ( v1_relat_1(G)
        & v1_funct_1(G) )
     => ! [H,I,J,K,L,M] :
          ( G = k1_funct_4(k1_funct_4(k1_funct_4(k1_funct_4(k1_funct_4(k3_cqc_lang(B,I),k3_cqc_lang(C,J)),k3_cqc_lang(D,K)),k3_cqc_lang(E,L)),k3_cqc_lang(F,M)),k3_cqc_lang(A,H))
         => k1_relat_1(G) = k4_enumset1(A,B,C,D,E,F) ) ) )).

fof(t9_bvfunc23,axiom,(
    ! [A,B,C,D,E,F,G] :
      ( ( v1_relat_1(G)
        & v1_funct_1(G) )
     => ! [H,I,J,K,L,M] :
          ( G = k1_funct_4(k1_funct_4(k1_funct_4(k1_funct_4(k1_funct_4(k3_cqc_lang(B,I),k3_cqc_lang(C,J)),k3_cqc_lang(D,K)),k3_cqc_lang(E,L)),k3_cqc_lang(F,M)),k3_cqc_lang(A,H))
         => ( A = B
            | A = C
            | A = D
            | A = E
            | A = F
            | B = C
            | B = D
            | B = E
            | B = F
            | C = D
            | C = E
            | C = F
            | D = E
            | D = F
            | E = F
            | k2_relat_1(G) = k4_enumset1(k1_funct_1(G,A),k1_funct_1(G,B),k1_funct_1(G,C),k1_funct_1(G,D),k1_funct_1(G,E),k1_funct_1(G,F)) ) ) ) )).

fof(t10_bvfunc23,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A)))
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => ! [D] :
                  ( m1_eqrel_1(D,A)
                 => ! [E] :
                      ( m1_eqrel_1(E,A)
                     => ! [F] :
                          ( m1_eqrel_1(F,A)
                         => ! [G] :
                              ( m1_eqrel_1(G,A)
                             => ! [H] :
                                  ( m1_eqrel_1(H,A)
                                 => ! [I] :
                                      ( m1_subset_1(I,A)
                                     => ! [J] :
                                          ( m1_subset_1(J,A)
                                         => ! [K] :
                                              ( ( v1_relat_1(K)
                                                & v1_funct_1(K) )
                                             => ~ ( v2_bvfunc_2(B,A)
                                                  & B = k4_enumset1(C,D,E,F,G,H)
                                                  & C != D
                                                  & C != E
                                                  & C != F
                                                  & C != G
                                                  & C != H
                                                  & D != E
                                                  & D != F
                                                  & D != G
                                                  & D != H
                                                  & E != F
                                                  & E != G
                                                  & E != H
                                                  & F != G
                                                  & F != H
                                                  & G != H
                                                  & r1_xboole_0(k22_bvfunc_1(A,J,k2_partit1(A,k2_partit1(A,k2_partit1(A,k2_partit1(A,D,E),F),G),H)),k22_bvfunc_1(A,I,C)) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t11_bvfunc23,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A)))
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => ! [D] :
                  ( m1_eqrel_1(D,A)
                 => ! [E] :
                      ( m1_eqrel_1(E,A)
                     => ! [F] :
                          ( m1_eqrel_1(F,A)
                         => ! [G] :
                              ( m1_eqrel_1(G,A)
                             => ! [H] :
                                  ( m1_eqrel_1(H,A)
                                 => ! [I] :
                                      ( m1_subset_1(I,A)
                                     => ! [J] :
                                          ( m1_subset_1(J,A)
                                         => ! [K] :
                                              ( ( v1_relat_1(K)
                                                & v1_funct_1(K) )
                                             => ~ ( v2_bvfunc_2(B,A)
                                                  & B = k4_enumset1(C,D,E,F,G,H)
                                                  & C != D
                                                  & C != E
                                                  & C != F
                                                  & C != G
                                                  & C != H
                                                  & D != E
                                                  & D != F
                                                  & D != G
                                                  & D != H
                                                  & E != F
                                                  & E != G
                                                  & E != H
                                                  & F != G
                                                  & F != H
                                                  & G != H
                                                  & k22_bvfunc_1(A,I,k2_partit1(A,k2_partit1(A,k2_partit1(A,E,F),G),H)) = k22_bvfunc_1(A,J,k2_partit1(A,k2_partit1(A,k2_partit1(A,E,F),G),H))
                                                  & r1_xboole_0(k22_bvfunc_1(A,J,k5_bvfunc_2(A,C,B)),k22_bvfunc_1(A,I,k5_bvfunc_2(A,D,B))) ) ) ) ) ) ) ) ) ) ) ) ) )).
%------------------------------------------------------------------------------