SET007 Axioms: SET007+901.ax


%------------------------------------------------------------------------------
% File     : SET007+901 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Properties of First and Second Order Cutting of Binary Relations
% 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    : relset_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   93 (   4 unit)
%            Number of atoms       :  352 (  72 equality)
%            Maximal formula depth :   15 (   8 average)
%            Number of connectives :  280 (  21 ~  ;   1  |;  67  &)
%                                         (  22 <=>; 169 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   12 (   1 propositional; 0-3 arity)
%            Number of functors    :   44 (   1 constant; 0-6 arity)
%            Number of variables   :  378 (   0 singleton; 365 !;  13 ?)
%            Maximal term depth    :    6 (   2 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_relset_2,axiom,(
    ! [A,B] :
      ( r2_hidden(A,k3_pua2mss1(B))
    <=> ? [C] :
          ( A = k1_tarski(C)
          & r2_hidden(C,B) ) ) )).

fof(t2_relset_2,axiom,(
    ! [A] :
      ( A = k1_xboole_0
    <=> k3_pua2mss1(A) = k1_xboole_0 ) )).

fof(t3_relset_2,axiom,(
    ! [A,B] : k3_pua2mss1(k2_xboole_0(A,B)) = k2_xboole_0(k3_pua2mss1(A),k3_pua2mss1(B)) )).

fof(t4_relset_2,axiom,(
    ! [A,B] : k3_pua2mss1(k3_xboole_0(A,B)) = k3_xboole_0(k3_pua2mss1(A),k3_pua2mss1(B)) )).

fof(t5_relset_2,axiom,(
    ! [A,B] : k3_pua2mss1(k4_xboole_0(A,B)) = k4_xboole_0(k3_pua2mss1(A),k3_pua2mss1(B)) )).

fof(t6_relset_2,axiom,(
    ! [A,B] :
      ( r1_tarski(A,B)
    <=> r1_tarski(k3_pua2mss1(A),k3_pua2mss1(B)) ) )).

fof(t7_relset_2,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
         => r1_tarski(k5_subset_1(A,k8_setfam_1(A,B),k8_setfam_1(A,C)),k8_setfam_1(A,k1_relset_2(A,B,C))) ) ) )).

fof(t8_relset_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( v1_relat_1(B)
         => ! [C] :
              ( v1_relat_1(C)
             => r1_tarski(k5_relat_1(k3_xboole_0(A,B),C),k3_xboole_0(k5_relat_1(A,C),k5_relat_1(B,C))) ) ) ) )).

fof(d1_relset_2,axiom,(
    ! [A,B,C] :
      ( m2_relset_1(C,A,B)
     => ! [D] :
          ( m1_subset_1(D,A)
         => k2_relset_2(A,B,C,D) = k10_relset_1(A,B,C,k1_tarski(D)) ) ) )).

fof(t9_relset_2,axiom,(
    ! [A,B,C] :
      ( v1_relat_1(C)
     => ( r2_hidden(A,k9_relat_1(C,k1_tarski(B)))
      <=> r2_hidden(k4_tarski(B,A),C) ) ) )).

fof(t10_relset_2,axiom,(
    ! [A,B] :
      ( v1_relat_1(B)
     => ! [C] :
          ( v1_relat_1(C)
         => k9_relat_1(k2_xboole_0(B,C),k1_tarski(A)) = k2_xboole_0(k9_relat_1(B,k1_tarski(A)),k9_relat_1(C,k1_tarski(A))) ) ) )).

fof(t11_relset_2,axiom,(
    ! [A,B] :
      ( v1_relat_1(B)
     => ! [C] :
          ( v1_relat_1(C)
         => k9_relat_1(k3_xboole_0(B,C),k1_tarski(A)) = k3_xboole_0(k9_relat_1(B,k1_tarski(A)),k9_relat_1(C,k1_tarski(A))) ) ) )).

fof(t12_relset_2,axiom,(
    ! [A,B] :
      ( v1_relat_1(B)
     => ! [C] :
          ( v1_relat_1(C)
         => k9_relat_1(k4_xboole_0(B,C),k1_tarski(A)) = k4_xboole_0(k9_relat_1(B,k1_tarski(A)),k9_relat_1(C,k1_tarski(A))) ) ) )).

fof(t13_relset_2,axiom,(
    ! [A,B] :
      ( v1_relat_1(B)
     => ! [C] :
          ( v1_relat_1(C)
         => r1_tarski(k9_relat_1(k3_xboole_0(B,C),k3_pua2mss1(A)),k3_xboole_0(k9_relat_1(B,k3_pua2mss1(A)),k9_relat_1(C,k3_pua2mss1(A)))) ) ) )).

fof(d2_relset_2,axiom,(
    ! [A,B,C] :
      ( m2_relset_1(C,A,B)
     => ! [D] :
          ( m1_subset_1(D,A)
         => k3_relset_2(A,B,C,D) = k11_relset_1(A,B,C,k1_tarski(D)) ) ) )).

fof(d3_relset_2,axiom,(
    ! [A,B] :
      ( v1_relat_1(B)
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( C = k5_relset_2(A,B)
          <=> ( k1_relat_1(C) = k1_zfmisc_1(A)
              & ! [D] :
                  ( r1_tarski(D,A)
                 => k1_funct_1(C,D) = k9_relat_1(B,D) ) ) ) ) ) )).

fof(t19_relset_2,axiom,(
    ! [A,B,C,D] :
      ( m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(B,C)))
     => ( r2_hidden(A,k1_relat_1(k5_relset_2(B,D)))
       => k1_funct_1(k5_relset_2(B,D),A) = k9_relat_1(D,A) ) ) )).

fof(t20_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
     => r1_tarski(k2_relat_1(k5_relset_2(A,C)),k1_zfmisc_1(k2_relat_1(C))) ) )).

fof(t21_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
     => ( v1_funct_1(k5_relset_2(A,C))
        & v1_funct_2(k5_relset_2(A,C),k1_zfmisc_1(A),k1_zfmisc_1(k2_relat_1(C)))
        & m2_relset_1(k5_relset_2(A,C),k1_zfmisc_1(A),k1_zfmisc_1(k2_relat_1(C))) ) ) )).

fof(t22_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
     => r1_tarski(k5_setfam_1(B,k4_relset_2(k1_zfmisc_1(A),B,k6_relset_2(B,A,C),A)),k9_relat_1(C,k3_tarski(A))) ) )).

fof(d4_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => ! [D] :
          ( m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B)))
         => k7_relset_2(A,B,C,D) = k8_setfam_1(B,k4_relset_2(k1_zfmisc_1(A),B,k6_relset_2(B,A,D),k3_pua2mss1(C))) ) ) )).

fof(t23_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => ! [D] :
          ( m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B)))
         => ( k4_relset_2(k1_zfmisc_1(A),B,k6_relset_2(B,A,D),k3_pua2mss1(C)) = k1_xboole_0
          <=> C = k1_xboole_0 ) ) ) )).

fof(t24_relset_2,axiom,(
    ! [A,B,C,D] :
      ( m1_subset_1(D,k1_zfmisc_1(A))
     => ! [E] :
          ( m1_subset_1(E,k1_zfmisc_1(k2_zfmisc_1(A,B)))
         => ( r2_hidden(C,k8_relset_2(A,B,D,E))
           => ! [F] :
                ( r2_hidden(F,D)
               => r2_hidden(C,k9_relat_1(E,k1_tarski(F))) ) ) ) ) )).

fof(t25_relset_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B,C] :
          ( m1_subset_1(C,k1_zfmisc_1(B))
         => ! [D] :
              ( m1_subset_1(D,A)
             => ! [E] :
                  ( m1_subset_1(E,k1_zfmisc_1(k2_zfmisc_1(B,A)))
                 => ( r2_hidden(D,k8_relset_2(B,A,C,E))
                  <=> ! [F] :
                        ( r2_hidden(F,C)
                       => r2_hidden(D,k9_relat_1(E,k1_tarski(F))) ) ) ) ) ) ) )).

fof(t26_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => ! [D] :
          ( m1_subset_1(D,k1_zfmisc_1(A))
         => ! [E] :
              ( m1_subset_1(E,k1_zfmisc_1(k2_zfmisc_1(A,B)))
             => ( k4_relset_2(k1_zfmisc_1(A),B,k6_relset_2(B,A,E),k3_pua2mss1(C)) = k1_xboole_0
               => k8_relset_2(A,B,k4_subset_1(A,C,D),E) = k8_relset_2(A,B,D,E) ) ) ) ) )).

fof(t27_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => ! [D] :
          ( m1_subset_1(D,k1_zfmisc_1(A))
         => ! [E] :
              ( m1_subset_1(E,k1_zfmisc_1(k2_zfmisc_1(A,B)))
             => k8_relset_2(A,B,k4_subset_1(A,C,D),E) = k5_subset_1(B,k8_relset_2(A,B,C,E),k8_relset_2(A,B,D,E)) ) ) ) )).

fof(t29_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => ! [D] :
          ( m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B)))
         => ( C = k1_xboole_0
           => k8_relset_2(A,B,C,D) = B ) ) ) )).

fof(t30_relset_2,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( k5_setfam_1(A,B) = k1_xboole_0
      <=> ! [C] :
            ( r2_hidden(C,B)
           => C = k1_xboole_0 ) ) ) )).

fof(t32_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => ! [D] :
          ( m1_subset_1(D,k1_zfmisc_1(A))
         => ! [E] :
              ( m1_subset_1(E,k1_zfmisc_1(k2_zfmisc_1(A,B)))
             => ( r1_tarski(C,D)
               => r1_tarski(k8_relset_2(A,B,D,E),k8_relset_2(A,B,C,E)) ) ) ) ) )).

fof(t33_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => ! [D] :
          ( m1_subset_1(D,k1_zfmisc_1(A))
         => ! [E] :
              ( m1_subset_1(E,k1_zfmisc_1(k2_zfmisc_1(A,B)))
             => r1_tarski(k4_subset_1(B,k8_relset_2(A,B,C,E),k8_relset_2(A,B,D,E)),k8_relset_2(A,B,k5_subset_1(A,C,D),E)) ) ) ) )).

fof(t34_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => ! [D] :
          ( m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B)))
         => ! [E] :
              ( m1_subset_1(E,k1_zfmisc_1(k2_zfmisc_1(A,B)))
             => k8_relset_2(A,B,C,k5_subset_1(k2_zfmisc_1(A,B),D,E)) = k5_subset_1(B,k8_relset_2(A,B,C,D),k8_relset_2(A,B,C,E)) ) ) ) )).

fof(t37_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => ! [D] :
          ( m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B)))
         => ( D = k1_xboole_0
           => ( C = k1_xboole_0
              | k8_relset_2(A,B,C,D) = k1_xboole_0 ) ) ) ) )).

fof(t38_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => ! [D] :
          ( m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B)))
         => ( D = k2_zfmisc_1(A,B)
           => k8_relset_2(A,B,C,D) = B ) ) ) )).

fof(t40_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => ! [D] :
          ( m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B)))
         => ! [E] :
              ( m1_subset_1(E,k1_zfmisc_1(k2_zfmisc_1(A,B)))
             => ( r1_tarski(D,E)
               => r1_tarski(k8_relset_2(A,B,C,D),k8_relset_2(A,B,C,E)) ) ) ) ) )).

fof(t41_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => ! [D] :
          ( m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B)))
         => ! [E] :
              ( m1_subset_1(E,k1_zfmisc_1(k2_zfmisc_1(A,B)))
             => r1_tarski(k4_subset_1(B,k8_relset_2(A,B,C,D),k8_relset_2(A,B,C,E)),k8_relset_2(A,B,C,k4_subset_1(k2_zfmisc_1(A,B),D,E))) ) ) ) )).

fof(t42_relset_2,axiom,(
    ! [A,B,C,D,E] :
      ( m1_subset_1(E,k1_zfmisc_1(k2_zfmisc_1(C,D)))
     => ( r2_hidden(A,k9_relat_1(k3_subset_1(k2_zfmisc_1(C,D),E),k1_tarski(B)))
      <=> ( ~ r2_hidden(k4_tarski(B,A),E)
          & r2_hidden(B,C)
          & r2_hidden(A,D) ) ) ) )).

fof(t43_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => ! [D] :
          ( m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B)))
         => ( C != k1_xboole_0
           => r1_tarski(k8_relset_2(A,B,C,D),k9_relat_1(D,C)) ) ) ) )).

fof(t44_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
     => ! [D,E] :
          ( ~ ( ~ r1_xboole_0(D,k9_relat_1(k4_relat_1(C),E))
              & ! [F,G] :
                  ~ ( r2_hidden(F,D)
                    & r2_hidden(G,E)
                    & r2_hidden(F,k9_relat_1(k4_relat_1(C),k1_tarski(G))) ) )
          & ~ ( ? [F,G] :
                  ( r2_hidden(F,D)
                  & r2_hidden(G,E)
                  & r2_hidden(F,k9_relat_1(k4_relat_1(C),k1_tarski(G))) )
              & r1_xboole_0(D,k9_relat_1(k4_relat_1(C),E)) ) ) ) )).

fof(t45_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
     => ! [D,E] :
          ( ~ ( ? [F,G] :
                  ( r2_hidden(F,D)
                  & r2_hidden(G,E)
                  & r2_hidden(F,k9_relat_1(k4_relat_1(C),k1_tarski(G))) )
              & r1_xboole_0(E,k9_relat_1(C,D)) )
          & ~ ( ~ r1_xboole_0(E,k9_relat_1(C,D))
              & ! [F,G] :
                  ~ ( r2_hidden(F,D)
                    & r2_hidden(G,E)
                    & r2_hidden(F,k9_relat_1(k4_relat_1(C),k1_tarski(G))) ) ) ) ) )).

fof(t46_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => ! [D] :
          ( m1_subset_1(D,k1_zfmisc_1(B))
         => ! [E] :
              ( m1_subset_1(E,k1_zfmisc_1(k2_zfmisc_1(A,B)))
             => ( r1_xboole_0(C,k9_relat_1(k4_relat_1(E),D))
              <=> r1_xboole_0(D,k9_relat_1(E,C)) ) ) ) ) )).

fof(t47_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
     => ! [D] : k9_relat_1(C,D) = k9_relat_1(C,k3_xboole_0(D,k1_funct_5(C))) ) )).

fof(t48_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
     => ! [D] : k9_relat_1(k4_relat_1(C),D) = k9_relat_1(k4_relat_1(C),k3_xboole_0(D,k2_funct_5(C))) ) )).

fof(t49_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => ! [D] :
          ( m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B)))
         => k3_subset_1(B,k8_relset_2(A,B,C,D)) = k9_relat_1(k3_subset_1(k2_zfmisc_1(A,B),D),C) ) ) )).

fof(t50_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => ! [D] :
          ( m2_relset_1(D,A,B)
         => k3_subset_1(B,k10_relset_1(A,B,D,C)) = k8_relset_2(A,B,C,k3_subset_1(k2_zfmisc_1(A,B),D)) ) ) )).

fof(t51_relset_2,axiom,(
    ! [A,B,C] :
      ( m2_relset_1(C,B,A)
     => ( k1_funct_5(C) = k10_relset_1(A,B,k6_relset_1(B,A,C),A)
        & k2_funct_5(C) = k10_relset_1(B,A,C,B) ) ) )).

fof(t52_relset_2,axiom,(
    ! [A,B,C,D] :
      ( m2_relset_1(D,A,B)
     => ! [E] :
          ( m2_relset_1(E,B,C)
         => ( k1_funct_5(k9_relset_2(A,B,C,D,E)) = k10_relset_1(B,A,k6_relset_1(A,B,D),k1_funct_5(E))
            & r1_tarski(k1_funct_5(k9_relset_2(A,B,C,D,E)),k1_funct_5(D)) ) ) ) )).

fof(t53_relset_2,axiom,(
    ! [A,B,C,D] :
      ( m2_relset_1(D,A,B)
     => ! [E] :
          ( m2_relset_1(E,B,C)
         => ( k2_funct_5(k9_relset_2(A,B,C,D,E)) = k10_relset_1(B,C,E,k2_funct_5(D))
            & r1_tarski(k2_funct_5(k9_relset_2(A,B,C,D,E)),k2_funct_5(E)) ) ) ) )).

fof(t54_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => ! [D] :
          ( m2_relset_1(D,A,B)
         => ( r1_tarski(C,k1_funct_5(D))
          <=> r1_tarski(C,k10_relset_1(A,A,k9_relset_2(A,B,A,D,k6_relset_1(A,B,D)),C)) ) ) ) )).

fof(t55_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(B))
     => ! [D] :
          ( m2_relset_1(D,A,B)
         => ( r1_tarski(C,k2_funct_5(D))
          <=> r1_tarski(C,k10_relset_1(B,B,k9_relset_2(B,A,B,k6_relset_1(A,B,D),D),C)) ) ) ) )).

fof(t56_relset_2,axiom,(
    ! [A,B,C] :
      ( m2_relset_1(C,B,A)
     => ( k1_funct_5(C) = k10_relset_1(A,B,k6_relset_1(B,A,C),A)
        & k10_relset_1(A,B,k6_relset_1(B,A,C),k10_relset_1(B,A,C,B)) = k10_relset_1(A,B,k6_relset_1(B,A,C),k2_funct_5(C)) ) ) )).

fof(t57_relset_2,axiom,(
    ! [A,B,C] :
      ( m2_relset_1(C,B,A)
     => k10_relset_1(A,B,k6_relset_1(B,A,C),A) = k10_relset_1(B,B,k9_relset_2(B,A,B,C,k6_relset_1(B,A,C)),B) ) )).

fof(t58_relset_2,axiom,(
    ! [A,B,C] :
      ( m2_relset_1(C,A,B)
     => k10_relset_1(A,B,C,A) = k10_relset_1(B,B,k9_relset_2(B,A,B,k6_relset_1(A,B,C),C),B) ) )).

fof(t59_relset_2,axiom,(
    ! [A,B,C,D] :
      ( m1_subset_1(D,k1_zfmisc_1(A))
     => ! [E] :
          ( m2_relset_1(E,A,B)
         => ! [F] :
              ( m2_relset_1(F,B,C)
             => k8_relset_2(B,C,k10_relset_1(A,B,E,D),F) = k8_relset_2(A,C,D,k3_subset_1(k2_zfmisc_1(A,C),k9_relset_2(A,B,C,E,k3_subset_1(k2_zfmisc_1(B,C),F)))) ) ) ) )).

fof(t60_relset_2,axiom,(
    ! [A,B,C] :
      ( m2_relset_1(C,A,B)
     => k4_relat_1(k3_subset_1(k2_zfmisc_1(A,B),C)) = k3_subset_1(k2_zfmisc_1(B,A),k6_relset_1(A,B,C)) ) )).

fof(t61_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => ! [D] :
          ( m1_subset_1(D,k1_zfmisc_1(B))
         => ! [E] :
              ( m2_relset_1(E,A,B)
             => ( r1_tarski(C,k8_relset_2(B,A,D,k6_relset_1(A,B,E)))
              <=> r1_tarski(D,k8_relset_2(A,B,C,E)) ) ) ) ) )).

fof(t62_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => ! [D] :
          ( m1_subset_1(D,k1_zfmisc_1(B))
         => ! [E] :
              ( m2_relset_1(E,A,B)
             => ( r1_tarski(k10_relset_1(A,B,E,k3_subset_1(A,C)),k3_subset_1(B,D))
              <=> r1_tarski(k10_relset_1(B,A,k6_relset_1(A,B,E),D),C) ) ) ) ) )).

fof(t63_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => ! [D] :
          ( m1_subset_1(D,k1_zfmisc_1(B))
         => ! [E] :
              ( m2_relset_1(E,A,B)
             => ( r1_tarski(C,k8_relset_2(B,A,k8_relset_2(A,B,C,E),k6_relset_1(A,B,E)))
                & r1_tarski(D,k8_relset_2(A,B,k8_relset_2(B,A,D,k6_relset_1(A,B,E)),E)) ) ) ) ) )).

fof(t64_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => ! [D] :
          ( m1_subset_1(D,k1_zfmisc_1(B))
         => ! [E] :
              ( m2_relset_1(E,A,B)
             => ( k8_relset_2(A,B,C,E) = k8_relset_2(A,B,k8_relset_2(B,A,k8_relset_2(A,B,C,E),k6_relset_1(A,B,E)),E)
                & k8_relset_2(B,A,D,k6_relset_1(A,B,E)) = k8_relset_2(B,A,k8_relset_2(A,B,k8_relset_2(B,A,D,k6_relset_1(A,B,E)),E),k6_relset_1(A,B,E)) ) ) ) ) )).

fof(t65_relset_2,axiom,(
    ! [A,B,C] :
      ( m2_relset_1(C,A,B)
     => k9_relset_2(A,A,B,k6_partfun1(A),C) = k9_relset_2(A,B,B,C,k6_partfun1(B)) ) )).

fof(dt_k1_relset_2,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
        & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
     => m1_subset_1(k1_relset_2(A,B,C),k1_zfmisc_1(k1_zfmisc_1(A))) ) )).

fof(commutativity_k1_relset_2,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
        & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
     => k1_relset_2(A,B,C) = k1_relset_2(A,C,B) ) )).

fof(idempotence_k1_relset_2,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
        & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
     => k1_relset_2(A,B,B) = B ) )).

fof(redefinition_k1_relset_2,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
        & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
     => k1_relset_2(A,B,C) = k3_xboole_0(B,C) ) )).

fof(dt_k2_relset_2,axiom,(
    ! [A,B,C,D] :
      ( ( m1_relset_1(C,A,B)
        & m1_subset_1(D,A) )
     => m1_subset_1(k2_relset_2(A,B,C,D),k1_zfmisc_1(B)) ) )).

fof(dt_k3_relset_2,axiom,(
    ! [A,B,C,D] :
      ( ( m1_relset_1(C,A,B)
        & m1_subset_1(D,A) )
     => m1_subset_1(k3_relset_2(A,B,C,D),k1_zfmisc_1(A)) ) )).

fof(dt_k4_relset_2,axiom,(
    ! [A,B,C,D] :
      ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,k1_zfmisc_1(B))))
     => m1_subset_1(k4_relset_2(A,B,C,D),k1_zfmisc_1(k1_zfmisc_1(B))) ) )).

fof(redefinition_k4_relset_2,axiom,(
    ! [A,B,C,D] :
      ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,k1_zfmisc_1(B))))
     => k4_relset_2(A,B,C,D) = k9_relat_1(C,D) ) )).

fof(dt_k5_relset_2,axiom,(
    ! [A,B] :
      ( v1_relat_1(B)
     => ( v1_relat_1(k5_relset_2(A,B))
        & v1_funct_1(k5_relset_2(A,B)) ) ) )).

fof(dt_k6_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(B,A)))
     => ( v1_funct_1(k6_relset_2(A,B,C))
        & v1_funct_2(k6_relset_2(A,B,C),k1_zfmisc_1(B),k1_zfmisc_1(A))
        & m2_relset_1(k6_relset_2(A,B,C),k1_zfmisc_1(B),k1_zfmisc_1(A)) ) ) )).

fof(redefinition_k6_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(B,A)))
     => k6_relset_2(A,B,C) = k5_relset_2(B,C) ) )).

fof(dt_k7_relset_2,axiom,(
    $true )).

fof(dt_k8_relset_2,axiom,(
    ! [A,B,C,D] :
      ( ( m1_subset_1(C,k1_zfmisc_1(A))
        & m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B))) )
     => m1_subset_1(k8_relset_2(A,B,C,D),k1_zfmisc_1(B)) ) )).

fof(redefinition_k8_relset_2,axiom,(
    ! [A,B,C,D] :
      ( ( m1_subset_1(C,k1_zfmisc_1(A))
        & m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B))) )
     => k8_relset_2(A,B,C,D) = k7_relset_2(A,B,C,D) ) )).

fof(dt_k9_relset_2,axiom,(
    ! [A,B,C,D,E] :
      ( ( m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B)))
        & m1_subset_1(E,k1_zfmisc_1(k2_zfmisc_1(B,C))) )
     => m2_relset_1(k9_relset_2(A,B,C,D,E),A,C) ) )).

fof(redefinition_k9_relset_2,axiom,(
    ! [A,B,C,D,E] :
      ( ( m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B)))
        & m1_subset_1(E,k1_zfmisc_1(k2_zfmisc_1(B,C))) )
     => k9_relset_2(A,B,C,D,E) = k5_relat_1(D,E) ) )).

fof(t14_relset_2,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ! [C] :
          ( v1_relat_1(C)
         => k9_relat_1(C,k5_setfam_1(A,B)) = k3_tarski(a_3_0_relset_2(A,B,C)) ) ) )).

fof(t15_relset_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => B = k3_tarski(a_2_0_relset_2(A,B)) ) ) )).

fof(t16_relset_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => m1_subset_1(a_2_0_relset_2(A,B),k1_zfmisc_1(k1_zfmisc_1(A))) ) ) )).

fof(t17_relset_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B,C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ! [D] :
              ( m2_relset_1(D,A,B)
             => k10_relset_1(A,B,D,C) = k3_tarski(a_4_0_relset_2(A,B,C,D)) ) ) ) )).

fof(t18_relset_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B,C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ! [D] :
              ( m2_relset_1(D,A,B)
             => m1_subset_1(a_4_0_relset_2(A,B,C,D),k1_zfmisc_1(k1_zfmisc_1(B))) ) ) ) )).

fof(t28_relset_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B,C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
         => ! [D] :
              ( m2_relset_1(D,A,B)
             => m1_subset_1(a_4_1_relset_2(A,B,C,D),k1_zfmisc_1(k1_zfmisc_1(B))) ) ) ) )).

fof(t31_relset_2,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( m2_relset_1(C,A,B)
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(A)))
             => ! [E] :
                  ( m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(B)))
                 => ( E = a_4_2_relset_2(A,B,C,D)
                   => k8_relset_2(A,B,k5_setfam_1(A,D),C) = k8_setfam_1(B,E) ) ) ) ) ) )).

fof(t35_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => ! [D] :
          ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(k2_zfmisc_1(A,B))))
         => k9_relat_1(k5_setfam_1(k2_zfmisc_1(A,B),D),C) = k3_tarski(a_4_3_relset_2(A,B,C,D)) ) ) )).

fof(t36_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(k2_zfmisc_1(A,B))))
     => ! [D,E,F] :
          ( m1_subset_1(F,k1_zfmisc_1(D))
         => m1_subset_1(a_6_0_relset_2(A,B,C,D,E,F),k1_zfmisc_1(k1_zfmisc_1(E))) ) ) )).

fof(t39_relset_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(B))
     => ! [D] :
          ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(k2_zfmisc_1(B,A))))
         => ! [E] :
              ( m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(A)))
             => ( E = a_4_4_relset_2(A,B,C,D)
               => k8_relset_2(B,A,C,k8_setfam_1(k2_zfmisc_1(B,A),D)) = k8_setfam_1(A,E) ) ) ) ) )).

fof(fraenkel_a_3_0_relset_2,axiom,(
    ! [A,B,C,D] :
      ( ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(B)))
        & v1_relat_1(D) )
     => ( r2_hidden(A,a_3_0_relset_2(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,k1_zfmisc_1(B))
            & A = k9_relat_1(D,E)
            & r2_hidden(E,C) ) ) ) )).

fof(fraenkel_a_2_0_relset_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & m1_subset_1(C,k1_zfmisc_1(B)) )
     => ( r2_hidden(A,a_2_0_relset_2(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,B)
            & A = k1_tarski(D)
            & r2_hidden(D,C) ) ) ) )).

fof(fraenkel_a_4_0_relset_2,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(B)
        & m1_subset_1(D,k1_zfmisc_1(B))
        & m2_relset_1(E,B,C) )
     => ( r2_hidden(A,a_4_0_relset_2(B,C,D,E))
      <=> ? [F] :
            ( m1_subset_1(F,B)
            & A = k2_relset_2(B,C,E,F)
            & r2_hidden(F,D) ) ) ) )).

fof(fraenkel_a_4_1_relset_2,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(B)
        & m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(B)))
        & m2_relset_1(E,B,C) )
     => ( r2_hidden(A,a_4_1_relset_2(B,C,D,E))
      <=> ? [F] :
            ( m1_subset_1(F,k1_zfmisc_1(B))
            & A = k8_relset_2(B,C,F,E)
            & r2_hidden(F,D) ) ) ) )).

fof(fraenkel_a_4_2_relset_2,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(C)
        & m2_relset_1(D,B,C)
        & m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(B))) )
     => ( r2_hidden(A,a_4_2_relset_2(B,C,D,E))
      <=> ? [F] :
            ( m1_subset_1(F,k1_zfmisc_1(B))
            & A = k8_relset_2(B,C,F,D)
            & r2_hidden(F,E) ) ) ) )).

fof(fraenkel_a_4_3_relset_2,axiom,(
    ! [A,B,C,D,E] :
      ( ( m1_subset_1(D,k1_zfmisc_1(B))
        & m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(k2_zfmisc_1(B,C)))) )
     => ( r2_hidden(A,a_4_3_relset_2(B,C,D,E))
      <=> ? [F] :
            ( m1_subset_1(F,k1_zfmisc_1(k2_zfmisc_1(B,C)))
            & A = k9_relat_1(F,D)
            & r2_hidden(F,E) ) ) ) )).

fof(fraenkel_a_6_0_relset_2,axiom,(
    ! [A,B,C,D,E,F,G] :
      ( ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(k2_zfmisc_1(B,C))))
        & m1_subset_1(G,k1_zfmisc_1(E)) )
     => ( r2_hidden(A,a_6_0_relset_2(B,C,D,E,F,G))
      <=> ? [H] :
            ( m1_subset_1(H,k1_zfmisc_1(k2_zfmisc_1(E,F)))
            & A = k8_relset_2(E,F,G,H)
            & r2_hidden(H,D) ) ) ) )).

fof(fraenkel_a_4_4_relset_2,axiom,(
    ! [A,B,C,D,E] :
      ( ( m1_subset_1(D,k1_zfmisc_1(C))
        & m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(k2_zfmisc_1(C,B)))) )
     => ( r2_hidden(A,a_4_4_relset_2(B,C,D,E))
      <=> ? [F] :
            ( m1_subset_1(F,k1_zfmisc_1(k2_zfmisc_1(C,B)))
            & A = k8_relset_2(C,B,D,F)
            & r2_hidden(F,E) ) ) ) )).
%------------------------------------------------------------------------------