SET007 Axioms: SET007+68.ax


%------------------------------------------------------------------------------
% File     : SET007+68 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Universal Classes
% 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    : classes2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  134 (  22 unit)
%            Number of atoms       :  515 (  56 equality)
%            Maximal formula depth :   17 (   5 average)
%            Number of connectives :  477 (  96 ~  ;   5  |; 229  &)
%                                         (   5 <=>; 142 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   21 (   1 propositional; 0-2 arity)
%            Number of functors    :   44 (   6 constant; 0-3 arity)
%            Number of variables   :  225 (   0 singleton; 222 !;   3 ?)
%            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(cc1_classes2,axiom,(
    ! [A] :
      ( v2_classes1(A)
     => v1_classes1(A) ) )).

fof(fc1_classes2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(k1_classes1(A))
      & v1_classes1(k1_classes1(A))
      & v2_classes1(k1_classes1(A)) ) )).

fof(cc2_classes2,axiom,(
    ! [A] :
      ( v1_classes2(A)
     => ( v1_ordinal1(A)
        & v1_classes1(A)
        & v2_classes1(A) ) ) )).

fof(cc3_classes2,axiom,(
    ! [A] :
      ( ( v1_ordinal1(A)
        & v2_classes1(A) )
     => v1_classes2(A) ) )).

fof(rc1_classes2,axiom,(
    ? [A] :
      ( v1_ordinal1(A)
      & ~ v1_xboole_0(A)
      & v1_classes1(A)
      & v2_classes1(A)
      & v1_classes2(A) ) )).

fof(fc2_classes2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ( v1_ordinal1(k2_ordinal2(A))
        & v2_ordinal1(k2_ordinal2(A))
        & v3_ordinal1(k2_ordinal2(A)) ) ) )).

fof(fc3_classes2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ( v1_ordinal1(k1_classes1(A))
        & ~ v1_xboole_0(k1_classes1(A))
        & v1_classes1(k1_classes1(A))
        & v2_classes1(k1_classes1(A))
        & v1_classes2(k1_classes1(A)) ) ) )).

fof(fc4_classes2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ( v1_ordinal1(k1_classes1(A))
        & ~ v1_xboole_0(k1_classes1(A))
        & v1_classes1(k1_classes1(A))
        & v2_classes1(k1_classes1(A))
        & v1_classes2(k1_classes1(A)) ) ) )).

fof(rc2_classes2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ? [B] :
          ( m1_subset_1(B,A)
          & ~ v1_xboole_0(B) ) ) )).

fof(fc5_classes2,axiom,(
    ! [A] : v1_ordinal1(k5_classes1(A)) )).

fof(fc6_classes2,axiom,(
    ! [A] :
      ( v1_ordinal1(A)
     => ( v1_ordinal1(k1_classes1(A))
        & ~ v1_xboole_0(k1_classes1(A))
        & v1_classes1(k1_classes1(A))
        & v2_classes1(k1_classes1(A))
        & v1_classes2(k1_classes1(A)) ) ) )).

fof(fc7_classes2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => v1_ordinal1(k4_classes1(A)) ) )).

fof(fc8_classes2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ( v1_ordinal1(k16_classes2(A))
        & ~ v1_xboole_0(k16_classes2(A))
        & v1_classes1(k16_classes2(A))
        & v2_classes1(k16_classes2(A))
        & v1_classes2(k16_classes2(A)) ) ) )).

fof(t1_classes2,axiom,(
    ! [A,B] :
      ( ( v1_classes1(A)
        & r2_hidden(B,A) )
     => ( ~ r2_wellord2(B,A)
        & r2_hidden(k1_card_1(B),k1_card_1(A)) ) ) )).

fof(t2_classes2,axiom,(
    $true )).

fof(t3_classes2,axiom,(
    ! [A,B,C] :
      ( ( v2_classes1(A)
        & r2_hidden(B,A)
        & r2_hidden(C,A) )
     => ( r2_hidden(k1_tarski(B),A)
        & r2_hidden(k2_tarski(B,C),A) ) ) )).

fof(t4_classes2,axiom,(
    ! [A,B,C] :
      ( ( v2_classes1(A)
        & r2_hidden(B,A)
        & r2_hidden(C,A) )
     => r2_hidden(k4_tarski(B,C),A) ) )).

fof(t5_classes2,axiom,(
    ! [A,B] :
      ( ( v2_classes1(A)
        & r2_hidden(B,A) )
     => r1_tarski(k1_classes1(B),A) ) )).

fof(t6_classes2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v2_classes1(B)
            & r2_hidden(A,B) )
         => ( r2_hidden(k1_ordinal1(A),B)
            & r1_tarski(A,B) ) ) ) )).

fof(t7_classes2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( r2_hidden(A,k1_classes1(B))
         => ( r2_hidden(k1_ordinal1(A),k1_classes1(B))
            & r1_tarski(A,k1_classes1(B)) ) ) ) )).

fof(t8_classes2,axiom,(
    ! [A,B] :
      ( ( v1_classes1(A)
        & v1_ordinal1(B)
        & r2_hidden(B,A) )
     => r1_tarski(B,A) ) )).

fof(t9_classes2,axiom,(
    ! [A,B] :
      ( ( v1_ordinal1(A)
        & r2_hidden(A,k1_classes1(B)) )
     => r1_tarski(A,k1_classes1(B)) ) )).

fof(t10_classes2,axiom,(
    ! [A] :
      ( v2_classes1(A)
     => k2_ordinal2(A) = k1_card_1(A) ) )).

fof(t11_classes2,axiom,(
    ! [A] : k2_ordinal2(k1_classes1(A)) = k1_card_1(k1_classes1(A)) )).

fof(t12_classes2,axiom,(
    ! [A,B] :
      ( ( v2_classes1(A)
        & r2_hidden(B,A) )
     => r2_hidden(k1_card_1(B),A) ) )).

fof(t13_classes2,axiom,(
    ! [A,B] :
      ( r2_hidden(A,k1_classes1(B))
     => r2_hidden(k1_card_1(A),k1_classes1(B)) ) )).

fof(t14_classes2,axiom,(
    ! [A,B] :
      ( ( v2_classes1(A)
        & r2_hidden(B,k1_card_1(A)) )
     => r2_hidden(B,A) ) )).

fof(t15_classes2,axiom,(
    ! [A,B] :
      ( r2_hidden(A,k1_card_1(k1_classes1(B)))
     => r2_hidden(A,k1_classes1(B)) ) )).

fof(t16_classes2,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( ( v2_classes1(B)
            & r2_hidden(A,k1_card_1(B)) )
         => r2_hidden(A,B) ) ) )).

fof(t17_classes2,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( r2_hidden(A,k1_card_1(k1_classes1(B)))
         => r2_hidden(A,k1_classes1(B)) ) ) )).

fof(t18_classes2,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( ( v2_classes1(B)
            & r2_hidden(A,B) )
         => r1_tarski(A,B) ) ) )).

fof(t19_classes2,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( r2_hidden(A,k1_classes1(B))
         => r1_tarski(A,k1_classes1(B)) ) ) )).

fof(t20_classes2,axiom,(
    ! [A] :
      ( v2_classes1(A)
     => v4_ordinal1(k1_card_1(A)) ) )).

fof(t21_classes2,axiom,(
    ! [A] :
      ( v2_classes1(A)
     => ( A = k1_xboole_0
        | ( k1_card_1(A) != np__0
          & k1_card_1(A) != k1_xboole_0
          & v4_ordinal1(k1_card_1(A)) ) ) ) )).

fof(t22_classes2,axiom,(
    ! [A] :
      ( k1_card_1(k1_classes1(A)) != np__0
      & k1_card_1(k1_classes1(A)) != k1_xboole_0
      & v4_ordinal1(k1_card_1(k1_classes1(A))) ) )).

fof(t23_classes2,axiom,(
    ! [A,B] :
      ( v2_classes1(A)
     => ( ( ~ ( r2_hidden(B,A)
              & v1_ordinal1(A) )
          & ~ ( r2_hidden(B,A)
              & r1_tarski(B,A) )
          & ~ ( r2_hidden(k1_card_1(B),k1_card_1(A))
              & r1_tarski(B,A) ) )
        | r1_tarski(k1_funct_2(B,A),A) ) ) )).

fof(t24_classes2,axiom,(
    ! [A,B] :
      ( ~ ( ~ ( r2_hidden(A,k1_classes1(B))
              & v1_ordinal1(B) )
          & ~ ( r2_hidden(A,k1_classes1(B))
              & r1_tarski(A,k1_classes1(B)) )
          & ~ ( r2_hidden(k1_card_1(A),k1_card_1(k1_classes1(B)))
              & r1_tarski(A,k1_classes1(B)) ) )
     => r1_tarski(k1_funct_2(A,k1_classes1(B)),k1_classes1(B)) ) )).

fof(t25_classes2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A) )
     => ( ( v4_ordinal1(k1_relat_1(A))
          & ! [B] :
              ( v3_ordinal1(B)
             => ( r2_hidden(B,k1_relat_1(A))
               => k1_funct_1(A,B) = k4_classes1(B) ) ) )
       => k4_classes1(k1_relat_1(A)) = k3_card_3(A) ) ) )).

fof(t26_classes2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v2_classes1(B)
            & r2_hidden(A,k2_ordinal2(B)) )
         => ( r2_hidden(k1_card_1(k4_classes1(A)),k1_card_1(B))
            & r2_hidden(k4_classes1(A),B) ) ) ) )).

fof(t27_classes2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( r2_hidden(A,k2_ordinal2(k1_classes1(B)))
         => ( r2_hidden(k1_card_1(k4_classes1(A)),k1_card_1(k1_classes1(B)))
            & r2_hidden(k4_classes1(A),k1_classes1(B)) ) ) ) )).

fof(t28_classes2,axiom,(
    ! [A] :
      ( v2_classes1(A)
     => r1_tarski(k4_classes1(k1_card_1(A)),A) ) )).

fof(t29_classes2,axiom,(
    ! [A] : r1_tarski(k4_classes1(k1_card_1(k1_classes1(A))),k1_classes1(A)) )).

fof(t30_classes2,axiom,(
    ! [A,B] :
      ( ( v2_classes1(A)
        & v1_ordinal1(A)
        & r2_hidden(B,A) )
     => r2_hidden(k6_classes1(B),A) ) )).

fof(t31_classes2,axiom,(
    ! [A] :
      ( ( v2_classes1(A)
        & v1_ordinal1(A) )
     => r1_tarski(A,k4_classes1(k1_card_1(A))) ) )).

fof(t32_classes2,axiom,(
    ! [A] :
      ( ( v2_classes1(A)
        & v1_ordinal1(A) )
     => k4_classes1(k1_card_1(A)) = A ) )).

fof(t33_classes2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v2_classes1(B)
            & r2_hidden(A,k2_ordinal2(B)) )
         => r1_tarski(k1_card_1(k4_classes1(A)),k1_card_1(B)) ) ) )).

fof(t34_classes2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( r2_hidden(A,k2_ordinal2(k1_classes1(B)))
         => r1_tarski(k1_card_1(k4_classes1(A)),k1_card_1(k1_classes1(B))) ) ) )).

fof(t35_classes2,axiom,(
    ! [A] :
      ( v2_classes1(A)
     => k1_card_1(A) = k1_card_1(k4_classes1(k1_card_1(A))) ) )).

fof(t36_classes2,axiom,(
    ! [A] : k1_card_1(k1_classes1(A)) = k1_card_1(k4_classes1(k1_card_1(k1_classes1(A)))) )).

fof(t37_classes2,axiom,(
    ! [A,B] :
      ~ ( v2_classes1(A)
        & r1_tarski(B,k4_classes1(k1_card_1(A)))
        & ~ r2_wellord2(B,k4_classes1(k1_card_1(A)))
        & ~ r2_hidden(B,k4_classes1(k1_card_1(A))) ) )).

fof(t38_classes2,axiom,(
    ! [A,B] :
      ~ ( r1_tarski(A,k4_classes1(k1_card_1(k1_classes1(B))))
        & ~ r2_wellord2(A,k4_classes1(k1_card_1(k1_classes1(B))))
        & ~ r2_hidden(A,k4_classes1(k1_card_1(k1_classes1(B)))) ) )).

fof(t39_classes2,axiom,(
    ! [A] :
      ( v2_classes1(A)
     => v2_classes1(k4_classes1(k1_card_1(A))) ) )).

fof(t40_classes2,axiom,(
    ! [A] : v2_classes1(k4_classes1(k1_card_1(k1_classes1(A)))) )).

fof(t41_classes2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ~ ( v1_ordinal1(B)
            & r2_hidden(A,k6_classes1(B))
            & ! [C] :
                ~ ( r2_hidden(C,B)
                  & k6_classes1(C) = A ) ) ) )).

fof(t42_classes2,axiom,(
    ! [A] :
      ( v1_ordinal1(A)
     => r1_tarski(k1_card_1(k6_classes1(A)),k1_card_1(A)) ) )).

fof(t43_classes2,axiom,(
    ! [A,B] :
      ( ( v2_classes1(A)
        & v1_ordinal1(B)
        & r2_hidden(B,A) )
     => r2_hidden(B,k4_classes1(k1_card_1(A))) ) )).

fof(t44_classes2,axiom,(
    ! [A,B] :
      ( ( v1_ordinal1(A)
        & r2_hidden(A,k1_classes1(B)) )
     => r2_hidden(A,k4_classes1(k1_card_1(k1_classes1(B)))) ) )).

fof(t45_classes2,axiom,(
    ! [A] :
      ( v1_ordinal1(A)
     => r1_classes1(A,k4_classes1(k1_card_1(k1_classes1(A)))) ) )).

fof(t46_classes2,axiom,(
    ! [A] :
      ( v1_ordinal1(A)
     => k4_classes1(k1_card_1(k1_classes1(A))) = k1_classes1(A) ) )).

fof(d1_classes2,axiom,(
    ! [A] :
      ( v1_classes2(A)
    <=> ( v1_ordinal1(A)
        & v2_classes1(A) ) ) )).

fof(t47_classes2,axiom,(
    $true )).

fof(t48_classes2,axiom,(
    $true )).

fof(t49_classes2,axiom,(
    $true )).

fof(t50_classes2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => v3_ordinal1(k2_ordinal2(A)) ) )).

fof(t51_classes2,axiom,(
    ! [A] :
      ( v1_ordinal1(A)
     => v1_classes2(k1_classes1(A)) ) )).

fof(t52_classes2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ( ~ v1_xboole_0(k1_classes1(A))
        & v1_classes2(k1_classes1(A)) ) ) )).

fof(t53_classes2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => v1_classes2(k1_classes1(A)) ) )).

fof(t54_classes2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => A = k4_classes1(k2_ordinal2(A)) ) )).

fof(t55_classes2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ( k2_ordinal2(A) != k1_xboole_0
        & v4_ordinal1(k2_ordinal2(A)) ) ) )).

fof(t56_classes2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_classes2(B) )
         => ~ ( ~ r2_hidden(A,B)
              & A != B
              & ~ r2_hidden(B,A) ) ) ) )).

fof(t57_classes2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_classes2(B) )
         => ( r1_tarski(A,B)
            | r2_hidden(B,A) ) ) ) )).

fof(t58_classes2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_classes2(B) )
         => r3_xboole_0(A,B) ) ) )).

fof(t59_classes2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_classes2(B) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v1_classes2(C) )
             => ( ( r2_hidden(A,B)
                  & r2_hidden(B,C) )
               => r2_hidden(A,C) ) ) ) ) )).

fof(t60_classes2,axiom,(
    $true )).

fof(t61_classes2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_classes2(B) )
         => ( ~ v1_xboole_0(k2_xboole_0(A,B))
            & v1_classes2(k2_xboole_0(A,B))
            & ~ v1_xboole_0(k3_xboole_0(A,B))
            & v1_classes2(k3_xboole_0(A,B)) ) ) ) )).

fof(t62_classes2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => r2_hidden(k1_xboole_0,A) ) )).

fof(t63_classes2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & v1_classes2(B) )
     => ( r2_hidden(A,B)
       => r2_hidden(k1_tarski(A),B) ) ) )).

fof(t64_classes2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(C)
        & v1_classes2(C) )
     => ( ( r2_hidden(A,C)
          & r2_hidden(B,C) )
       => ( r2_hidden(k2_tarski(A,B),C)
          & r2_hidden(k4_tarski(A,B),C) ) ) ) )).

fof(t65_classes2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & v1_classes2(B) )
     => ( r2_hidden(A,B)
       => ( r2_hidden(k1_zfmisc_1(A),B)
          & r2_hidden(k3_tarski(A),B)
          & r2_hidden(k1_setfam_1(A),B) ) ) ) )).

fof(t66_classes2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(C)
        & v1_classes2(C) )
     => ( ( r2_hidden(A,C)
          & r2_hidden(B,C) )
       => ( r2_hidden(k2_xboole_0(A,B),C)
          & r2_hidden(k3_xboole_0(A,B),C)
          & r2_hidden(k4_xboole_0(A,B),C)
          & r2_hidden(k5_xboole_0(A,B),C) ) ) ) )).

fof(t67_classes2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(C)
        & v1_classes2(C) )
     => ( ( r2_hidden(A,C)
          & r2_hidden(B,C) )
       => ( r2_hidden(k2_zfmisc_1(A,B),C)
          & r2_hidden(k1_funct_2(A,B),C) ) ) ) )).

fof(d2_classes2,axiom,(
    k13_classes2 = k1_classes1(k1_xboole_0) )).

fof(t68_classes2,axiom,(
    $true )).

fof(t69_classes2,axiom,(
    k1_card_1(k4_classes1(k5_ordinal2)) = k1_card_1(k5_ordinal2) )).

fof(t70_classes2,axiom,(
    v2_classes1(k4_classes1(k5_ordinal2)) )).

fof(t71_classes2,axiom,(
    k13_classes2 = k4_classes1(k5_ordinal2) )).

fof(d3_classes2,axiom,(
    k14_classes2 = k1_classes1(k13_classes2) )).

fof(d4_classes2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & v1_classes2(B) )
     => ( B = k15_classes2(A)
      <=> ( r1_tarski(A,B)
          & ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v1_classes2(C) )
             => ( r1_tarski(A,C)
               => r1_tarski(B,C) ) ) ) ) ) )).

fof(d5_classes2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( B = k16_classes2(A)
        <=> ? [C] :
              ( v1_relat_1(C)
              & v1_funct_1(C)
              & v5_ordinal1(C)
              & B = k1_ordinal2(C)
              & k1_relat_1(C) = k1_ordinal1(A)
              & k1_funct_1(C,k1_xboole_0) = k13_classes2
              & ! [D] :
                  ( v3_ordinal1(D)
                 => ( r2_hidden(k1_ordinal1(D),k1_ordinal1(A))
                   => k1_funct_1(C,k1_ordinal1(D)) = k1_classes1(k1_funct_1(C,D)) ) )
              & ! [D] :
                  ( v3_ordinal1(D)
                 => ( ( r2_hidden(D,k1_ordinal1(A))
                      & v4_ordinal1(D) )
                   => ( D = k1_xboole_0
                      | k1_funct_1(C,D) = k15_classes2(k3_card_3(k2_ordinal1(C,D))) ) ) ) ) ) ) )).

fof(t72_classes2,axiom,(
    $true )).

fof(t73_classes2,axiom,(
    $true )).

fof(t74_classes2,axiom,(
    $true )).

fof(t75_classes2,axiom,(
    k16_classes2(k1_xboole_0) = k13_classes2 )).

fof(t76_classes2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => k16_classes2(k1_ordinal1(A)) = k1_classes1(k16_classes2(A)) ) )).

fof(t77_classes2,axiom,(
    k16_classes2(k4_ordinal2) = k14_classes2 )).

fof(t78_classes2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B) )
         => ( ( v4_ordinal1(A)
              & k1_relat_1(B) = A
              & ! [C] :
                  ( v3_ordinal1(C)
                 => ( r2_hidden(C,A)
                   => k1_funct_1(B,C) = k16_classes2(C) ) ) )
           => ( A = k1_xboole_0
              | k16_classes2(A) = k15_classes2(k3_card_3(B)) ) ) ) ) )).

fof(t79_classes2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ( r1_tarski(k13_classes2,A)
        & r1_tarski(k1_classes1(k1_xboole_0),A)
        & r1_tarski(k16_classes2(k1_xboole_0),A) ) ) )).

fof(t80_classes2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ( r2_hidden(A,B)
          <=> r2_hidden(k16_classes2(A),k16_classes2(B)) ) ) ) )).

fof(t81_classes2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ( k16_classes2(A) = k16_classes2(B)
           => A = B ) ) ) )).

fof(t82_classes2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ( r1_ordinal1(A,B)
          <=> r1_tarski(k16_classes2(A),k16_classes2(B)) ) ) ) )).

fof(dt_k1_classes2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_subset_1(B,A) )
     => m1_subset_1(k1_classes2(A,B),A) ) )).

fof(redefinition_k1_classes2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_subset_1(B,A) )
     => k1_classes2(A,B) = k1_tarski(B) ) )).

fof(dt_k2_classes2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_subset_1(B,A) )
     => ( ~ v1_xboole_0(k2_classes2(A,B))
        & m1_subset_1(k2_classes2(A,B),A) ) ) )).

fof(redefinition_k2_classes2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_subset_1(B,A) )
     => k2_classes2(A,B) = k1_zfmisc_1(B) ) )).

fof(dt_k3_classes2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_subset_1(B,A) )
     => m1_subset_1(k3_classes2(A,B),A) ) )).

fof(redefinition_k3_classes2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_subset_1(B,A) )
     => k3_classes2(A,B) = k3_tarski(B) ) )).

fof(dt_k4_classes2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_subset_1(B,A) )
     => m1_subset_1(k4_classes2(A,B),A) ) )).

fof(redefinition_k4_classes2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_subset_1(B,A) )
     => k4_classes2(A,B) = k1_setfam_1(B) ) )).

fof(dt_k5_classes2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => m1_subset_1(k5_classes2(A,B,C),A) ) )).

fof(commutativity_k5_classes2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k5_classes2(A,B,C) = k5_classes2(A,C,B) ) )).

fof(redefinition_k5_classes2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k5_classes2(A,B,C) = k2_tarski(B,C) ) )).

fof(dt_k6_classes2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => m1_subset_1(k6_classes2(A,B,C),A) ) )).

fof(redefinition_k6_classes2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k6_classes2(A,B,C) = k4_tarski(B,C) ) )).

fof(dt_k7_classes2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => m1_subset_1(k7_classes2(A,B,C),A) ) )).

fof(commutativity_k7_classes2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k7_classes2(A,B,C) = k7_classes2(A,C,B) ) )).

fof(idempotence_k7_classes2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k7_classes2(A,B,B) = B ) )).

fof(redefinition_k7_classes2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k7_classes2(A,B,C) = k2_xboole_0(B,C) ) )).

fof(dt_k8_classes2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => m1_subset_1(k8_classes2(A,B,C),A) ) )).

fof(commutativity_k8_classes2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k8_classes2(A,B,C) = k8_classes2(A,C,B) ) )).

fof(idempotence_k8_classes2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k8_classes2(A,B,B) = B ) )).

fof(redefinition_k8_classes2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k8_classes2(A,B,C) = k3_xboole_0(B,C) ) )).

fof(dt_k9_classes2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => m1_subset_1(k9_classes2(A,B,C),A) ) )).

fof(redefinition_k9_classes2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k9_classes2(A,B,C) = k4_xboole_0(B,C) ) )).

fof(dt_k10_classes2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => m1_subset_1(k10_classes2(A,B,C),A) ) )).

fof(commutativity_k10_classes2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k10_classes2(A,B,C) = k10_classes2(A,C,B) ) )).

fof(redefinition_k10_classes2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k10_classes2(A,B,C) = k5_xboole_0(B,C) ) )).

fof(dt_k11_classes2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => m1_subset_1(k11_classes2(A,B,C),A) ) )).

fof(redefinition_k11_classes2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k11_classes2(A,B,C) = k2_zfmisc_1(B,C) ) )).

fof(dt_k12_classes2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => m1_subset_1(k12_classes2(A,B,C),A) ) )).

fof(redefinition_k12_classes2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k12_classes2(A,B,C) = k1_funct_2(B,C) ) )).

fof(dt_k13_classes2,axiom,
    ( ~ v1_xboole_0(k13_classes2)
    & v1_classes2(k13_classes2) )).

fof(dt_k14_classes2,axiom,
    ( ~ v1_xboole_0(k14_classes2)
    & v1_classes2(k14_classes2) )).

fof(dt_k15_classes2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(k15_classes2(A))
      & v1_classes2(k15_classes2(A)) ) )).

fof(dt_k16_classes2,axiom,(
    $true )).
%------------------------------------------------------------------------------