SET007 Axioms: SET007+186.ax


%------------------------------------------------------------------------------
% File     : SET007+186 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Definitions and Properties of Many Sorted 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    : mboolean [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   43 (   8 unit)
%            Number of atoms       :  170 (   5 equality)
%            Maximal formula depth :   14 (   7 average)
%            Number of connectives :  130 (   3 ~  ;   2  |;  21  &)
%                                         (   8 <=>;  96 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   14 (   1 propositional; 0-3 arity)
%            Number of functors    :   16 (   1 constant; 0-3 arity)
%            Number of variables   :  128 (   0 singleton; 127 !;   1 ?)
%            Maximal term depth    :    5 (   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(fc1_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ( v1_relat_1(k1_mboolean(A,B))
        & v2_relat_1(k1_mboolean(A,B))
        & v1_funct_1(k1_mboolean(A,B)) ) ) )).

fof(fc2_mboolean,axiom,(
    ! [A] :
      ( v1_relat_1(k2_mboolean(A,k1_pboole(A)))
      & v3_relat_1(k2_mboolean(A,k1_pboole(A)))
      & v1_funct_1(k2_mboolean(A,k1_pboole(A))) ) )).

fof(d1_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => ( C = k1_mboolean(A,B)
          <=> ! [D] :
                ( r2_hidden(D,A)
               => k1_funct_1(C,D) = k1_zfmisc_1(k1_funct_1(B,D)) ) ) ) ) )).

fof(t1_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => ( r6_pboole(A,B,k1_mboolean(A,C))
          <=> ! [D] :
                ( m1_pboole(D,A)
               => ( r1_pboole(A,D,B)
                <=> r2_pboole(A,D,C) ) ) ) ) ) )).

fof(t2_mboolean,axiom,(
    ! [A] : r6_pboole(A,k1_mboolean(A,k1_pboole(A)),k2_pre_circ(A,k1_tarski(k1_xboole_0))) )).

fof(t3_mboolean,axiom,(
    ! [A,B] : r6_pboole(A,k1_mboolean(A,k2_pre_circ(A,B)),k2_pre_circ(A,k1_zfmisc_1(B))) )).

fof(t4_mboolean,axiom,(
    ! [A,B] : r6_pboole(A,k1_mboolean(A,k2_pre_circ(A,k1_tarski(B))),k2_pre_circ(A,k2_tarski(k1_xboole_0,k1_tarski(B)))) )).

fof(t5_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => r2_pboole(A,k1_pboole(A),B) ) )).

fof(t6_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => ( r2_pboole(A,B,C)
           => r2_pboole(A,k1_mboolean(A,B),k1_mboolean(A,C)) ) ) ) )).

fof(t7_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => r2_pboole(A,k2_pboole(A,k1_mboolean(A,B),k1_mboolean(A,C)),k1_mboolean(A,k2_pboole(A,B,C))) ) ) )).

fof(t8_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => ( r6_pboole(A,k2_pboole(A,k1_mboolean(A,B),k1_mboolean(A,C)),k1_mboolean(A,k2_pboole(A,B,C)))
           => ! [D] :
                ( r2_hidden(D,A)
               => r3_xboole_0(k1_funct_1(B,D),k1_funct_1(C,D)) ) ) ) ) )).

fof(t9_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => r6_pboole(A,k1_mboolean(A,k3_pboole(A,B,C)),k3_pboole(A,k1_mboolean(A,B),k1_mboolean(A,C))) ) ) )).

fof(t10_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => r2_pboole(A,k1_mboolean(A,k4_pboole(A,B,C)),k2_pboole(A,k2_pre_circ(A,k1_tarski(k1_xboole_0)),k4_pboole(A,k1_mboolean(A,B),k1_mboolean(A,C)))) ) ) )).

fof(t11_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => ! [D] :
              ( m1_pboole(D,A)
             => ( r2_pboole(A,B,k4_pboole(A,C,D))
              <=> ( r2_pboole(A,B,C)
                  & r5_pboole(A,B,D) ) ) ) ) ) )).

fof(t12_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => r2_pboole(A,k2_pboole(A,k1_mboolean(A,k4_pboole(A,B,C)),k1_mboolean(A,k4_pboole(A,C,B))),k1_mboolean(A,k5_pboole(A,B,C))) ) ) )).

fof(t13_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => ! [D] :
              ( m1_pboole(D,A)
             => ( r2_pboole(A,B,k5_pboole(A,C,D))
              <=> ( r2_pboole(A,B,k2_pboole(A,C,D))
                  & r5_pboole(A,B,k3_pboole(A,C,D)) ) ) ) ) ) )).

fof(t14_mboolean,axiom,(
    $true )).

fof(t15_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => ! [D] :
              ( m1_pboole(D,A)
             => ( ( r2_pboole(A,B,C)
                  | r2_pboole(A,D,C) )
               => r2_pboole(A,k3_pboole(A,B,D),C) ) ) ) ) )).

fof(t16_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => ! [D] :
              ( m1_pboole(D,A)
             => ( r2_pboole(A,B,C)
               => r2_pboole(A,k4_pboole(A,B,D),C) ) ) ) ) )).

fof(t17_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => ! [D] :
              ( m1_pboole(D,A)
             => ( ( r2_pboole(A,B,C)
                  & r2_pboole(A,D,C) )
               => r2_pboole(A,k5_pboole(A,B,D),C) ) ) ) ) )).

fof(t18_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => r2_pboole(A,k11_pboole(A,B,C),k1_mboolean(A,k1_mboolean(A,k2_pboole(A,B,C)))) ) ) )).

fof(t19_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => ( r2_pboole(A,B,C)
          <=> r1_pboole(A,B,k1_mboolean(A,C)) ) ) ) )).

fof(t20_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => r2_pboole(A,k12_pboole(A,B,C),k1_mboolean(A,k11_pboole(A,B,C))) ) ) )).

fof(d2_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => ( C = k2_mboolean(A,B)
          <=> ! [D] :
                ( r2_hidden(D,A)
               => k1_funct_1(C,D) = k3_tarski(k1_funct_1(B,D)) ) ) ) ) )).

fof(t21_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => ( r1_pboole(A,B,k2_mboolean(A,C))
          <=> ? [D] :
                ( m1_pboole(D,A)
                & r1_pboole(A,B,D)
                & r1_pboole(A,D,C) ) ) ) ) )).

fof(t22_mboolean,axiom,(
    ! [A] : r6_pboole(A,k2_mboolean(A,k1_pboole(A)),k1_pboole(A)) )).

fof(t23_mboolean,axiom,(
    ! [A,B] : r6_pboole(A,k2_mboolean(A,k2_pre_circ(A,B)),k2_pre_circ(A,k3_tarski(B))) )).

fof(t24_mboolean,axiom,(
    ! [A,B] : r6_pboole(A,k2_mboolean(A,k2_pre_circ(A,k1_tarski(B))),k2_pre_circ(A,B)) )).

fof(t25_mboolean,axiom,(
    ! [A,B,C] : r6_pboole(A,k2_mboolean(A,k2_pre_circ(A,k2_tarski(k1_tarski(B),k1_tarski(C)))),k2_pre_circ(A,k2_tarski(B,C))) )).

fof(t26_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => ( r1_pboole(A,B,C)
           => r2_pboole(A,B,k2_mboolean(A,C)) ) ) ) )).

fof(t27_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => ( r2_pboole(A,B,C)
           => r2_pboole(A,k2_mboolean(A,B),k2_mboolean(A,C)) ) ) ) )).

fof(t28_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => r6_pboole(A,k2_mboolean(A,k2_pboole(A,B,C)),k2_pboole(A,k2_mboolean(A,B),k2_mboolean(A,C))) ) ) )).

fof(t29_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => r2_pboole(A,k2_mboolean(A,k3_pboole(A,B,C)),k3_pboole(A,k2_mboolean(A,B),k2_mboolean(A,C))) ) ) )).

fof(t30_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => r6_pboole(A,k2_mboolean(A,k1_mboolean(A,B)),B) ) )).

fof(t31_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => r2_pboole(A,B,k1_mboolean(A,k2_mboolean(A,B))) ) )).

fof(t32_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => ! [D] :
              ( m1_pboole(D,A)
             => ( ( r2_pboole(A,k2_mboolean(A,B),C)
                  & r1_pboole(A,D,B) )
               => r2_pboole(A,D,C) ) ) ) ) )).

fof(t33_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( ( v2_relat_1(C)
            & m1_pboole(C,A) )
         => ( ! [D] :
                ( m1_pboole(D,A)
               => ( r1_pboole(A,D,C)
                 => r2_pboole(A,D,B) ) )
           => r2_pboole(A,k2_mboolean(A,C),B) ) ) ) )).

fof(t34_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( ( v2_relat_1(C)
            & m1_pboole(C,A) )
         => ( ! [D] :
                ( m1_pboole(D,A)
               => ( r1_pboole(A,D,C)
                 => r6_pboole(A,k3_pboole(A,D,B),k1_pboole(A)) ) )
           => r6_pboole(A,k3_pboole(A,k2_mboolean(A,C),B),k1_pboole(A)) ) ) ) )).

fof(t35_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => ( ( v2_relat_1(k2_pboole(A,B,C))
              & ! [D] :
                  ( m1_pboole(D,A)
                 => ! [E] :
                      ( m1_pboole(E,A)
                     => ( ( r1_pboole(A,D,k2_pboole(A,B,C))
                          & r1_pboole(A,E,k2_pboole(A,B,C)) )
                       => ( D = E
                          | r6_pboole(A,k3_pboole(A,D,E),k1_pboole(A)) ) ) ) ) )
           => r6_pboole(A,k2_mboolean(A,k3_pboole(A,B,C)),k3_pboole(A,k2_mboolean(A,B),k2_mboolean(A,C))) ) ) ) )).

fof(t36_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => ! [D] :
              ( ( v2_relat_1(D)
                & m1_pboole(D,A) )
             => ( ( r2_pboole(A,C,k2_mboolean(A,k2_pboole(A,B,D)))
                  & ! [E] :
                      ( m1_pboole(E,A)
                     => ( r1_pboole(A,E,D)
                       => r6_pboole(A,k3_pboole(A,E,C),k1_pboole(A)) ) ) )
               => r2_pboole(A,C,k2_mboolean(A,B)) ) ) ) ) )).

fof(t37_mboolean,axiom,(
    ! [A,B] :
      ( ( v2_relat_1(B)
        & v1_pre_circ(B,A)
        & m1_pboole(B,A) )
     => ( ! [C] :
            ( m1_pboole(C,A)
           => ! [D] :
                ( m1_pboole(D,A)
               => ~ ( r1_pboole(A,C,B)
                    & r1_pboole(A,D,B)
                    & ~ r2_pboole(A,C,D)
                    & ~ r2_pboole(A,D,C) ) ) )
       => r1_pboole(A,k2_mboolean(A,B),B) ) ) )).

fof(dt_k1_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => m1_pboole(k1_mboolean(A,B),A) ) )).

fof(dt_k2_mboolean,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => m1_pboole(k2_mboolean(A,B),A) ) )).
%------------------------------------------------------------------------------