SET007 Axioms: SET007+60.ax


%------------------------------------------------------------------------------
% File     : SET007+60 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Equivalence Relations and Classes of Abstraction
% 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    : eqrel_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   80 (  16 unit)
%            Number of atoms       :  453 (  51 equality)
%            Maximal formula depth :   21 (   7 average)
%            Number of connectives :  398 (  25 ~  ;   2  |; 261  &)
%                                         (  11 <=>;  99 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   24 (   1 propositional; 0-3 arity)
%            Number of functors    :   27 (   5 constant; 0-4 arity)
%            Number of variables   :  199 (   0 singleton; 191 !;   8 ?)
%            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(fc1_eqrel_1,axiom,(
    ! [A] :
      ( v1_relat_1(k1_eqrel_1(A))
      & v1_relat_2(k1_eqrel_1(A))
      & v1_partfun1(k1_eqrel_1(A),A,A) ) )).

fof(fc2_eqrel_1,axiom,(
    ! [A] :
      ( v1_relat_1(k1_eqrel_1(A))
      & v1_relat_2(k1_eqrel_1(A))
      & v3_relat_2(k1_eqrel_1(A))
      & v8_relat_2(k1_eqrel_1(A))
      & v1_partfun1(k1_eqrel_1(A),A,A) ) )).

fof(cc1_eqrel_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ~ v1_xboole_0(B) ) ) )).

fof(cc2_eqrel_1,axiom,(
    ! [A,B] :
      ( m1_eqrel_1(B,A)
     => v1_setfam_1(B) ) )).

fof(fc3_eqrel_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v3_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m1_relset_1(B,A,A) )
     => ~ v1_xboole_0(k7_eqrel_1(A,B)) ) )).

fof(fc4_eqrel_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v3_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m1_relset_1(B,A,A) )
     => ( ~ v1_xboole_0(k7_eqrel_1(A,B))
        & v1_setfam_1(k7_eqrel_1(A,B)) ) ) )).

fof(d1_eqrel_1,axiom,(
    ! [A] : k1_eqrel_1(A) = k2_zfmisc_1(A,A) )).

fof(t1_eqrel_1,axiom,(
    $true )).

fof(t2_eqrel_1,axiom,(
    $true )).

fof(t3_eqrel_1,axiom,(
    $true )).

fof(t4_eqrel_1,axiom,(
    ! [A] :
      ( r1_relat_2(k6_partfun1(A),A)
      & r3_relat_2(k6_partfun1(A),A)
      & r8_relat_2(k6_partfun1(A),A) ) )).

fof(t5_eqrel_1,axiom,(
    $true )).

fof(t6_eqrel_1,axiom,(
    ! [A] :
      ( v3_relat_2(k6_partfun1(A))
      & v8_relat_2(k6_partfun1(A))
      & v1_partfun1(k6_partfun1(A),A,A)
      & m2_relset_1(k6_partfun1(A),A,A) ) )).

fof(t7_eqrel_1,axiom,(
    ! [A] :
      ( v3_relat_2(k1_eqrel_1(A))
      & v8_relat_2(k1_eqrel_1(A))
      & v1_partfun1(k1_eqrel_1(A),A,A)
      & m2_relset_1(k1_eqrel_1(A),A,A) ) )).

fof(t8_eqrel_1,axiom,(
    $true )).

fof(t9_eqrel_1,axiom,(
    $true )).

fof(t10_eqrel_1,axiom,(
    $true )).

fof(t11_eqrel_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_2(C)
        & v1_partfun1(C,A,A)
        & m2_relset_1(C,A,A) )
     => ( r2_hidden(B,A)
       => r2_hidden(k4_tarski(B,B),C) ) ) )).

fof(t12_eqrel_1,axiom,(
    ! [A,B,C,D] :
      ( ( v3_relat_2(D)
        & v1_partfun1(D,A,A)
        & m2_relset_1(D,A,A) )
     => ( r2_hidden(k4_tarski(B,C),D)
       => r2_hidden(k4_tarski(C,B),D) ) ) )).

fof(t13_eqrel_1,axiom,(
    ! [A,B,C,D,E] :
      ( ( v8_relat_2(E)
        & v1_partfun1(E,A,A)
        & m2_relset_1(E,A,A) )
     => ( ( r2_hidden(k4_tarski(B,C),E)
          & r2_hidden(k4_tarski(C,D),E) )
       => r2_hidden(k4_tarski(B,D),E) ) ) )).

fof(t14_eqrel_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ~ ( ? [C] : r2_hidden(C,A)
          & B = k1_xboole_0 ) ) )).

fof(t15_eqrel_1,axiom,(
    $true )).

fof(t16_eqrel_1,axiom,(
    ! [A,B] :
      ( m2_relset_1(B,A,A)
     => ( ( v3_relat_2(B)
          & v8_relat_2(B)
          & v1_partfun1(B,A,A)
          & m2_relset_1(B,A,A) )
      <=> ( v1_relat_2(B)
          & v3_relat_2(B)
          & v8_relat_2(B)
          & k3_relat_1(B) = A ) ) ) )).

fof(t17_eqrel_1,axiom,(
    ! [A,B] :
      ( ( v3_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => k4_eqrel_1(A,k6_partfun1(A),B) = k6_partfun1(A) ) )).

fof(t18_eqrel_1,axiom,(
    ! [A,B] :
      ( m2_relset_1(B,A,A)
     => k2_eqrel_1(A,k1_eqrel_1(A),B) = B ) )).

fof(t19_eqrel_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(k2_zfmisc_1(A,A))))
     => ( ! [C] :
            ( r2_hidden(C,B)
           => ( v3_relat_2(C)
              & v8_relat_2(C)
              & v1_partfun1(C,A,A)
              & m2_relset_1(C,A,A) ) )
       => ( B = k1_xboole_0
          | ( v3_relat_2(k6_setfam_1(k2_zfmisc_1(A,A),B))
            & v8_relat_2(k6_setfam_1(k2_zfmisc_1(A,A),B))
            & v1_partfun1(k6_setfam_1(k2_zfmisc_1(A,A),B),A,A)
            & m2_relset_1(k6_setfam_1(k2_zfmisc_1(A,A),B),A,A) ) ) ) ) )).

fof(t20_eqrel_1,axiom,(
    ! [A,B] :
      ( m2_relset_1(B,A,A)
     => ? [C] :
          ( v3_relat_2(C)
          & v8_relat_2(C)
          & v1_partfun1(C,A,A)
          & m2_relset_1(C,A,A)
          & r1_tarski(B,C)
          & ! [D] :
              ( ( v3_relat_2(D)
                & v8_relat_2(D)
                & v1_partfun1(D,A,A)
                & m2_relset_1(D,A,A) )
             => ( r1_tarski(B,D)
               => r1_tarski(C,D) ) ) ) ) )).

fof(d2_eqrel_1,axiom,(
    $true )).

fof(d3_eqrel_1,axiom,(
    ! [A,B] :
      ( ( v3_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] :
          ( ( v3_relat_2(C)
            & v8_relat_2(C)
            & v1_partfun1(C,A,A)
            & m2_relset_1(C,A,A) )
         => ! [D] :
              ( ( v3_relat_2(D)
                & v8_relat_2(D)
                & v1_partfun1(D,A,A)
                & m2_relset_1(D,A,A) )
             => ( D = k5_eqrel_1(A,B,C)
              <=> ( r1_tarski(k3_eqrel_1(A,B,C),D)
                  & ! [E] :
                      ( ( v3_relat_2(E)
                        & v8_relat_2(E)
                        & v1_partfun1(E,A,A)
                        & m2_relset_1(E,A,A) )
                     => ( r1_tarski(k3_eqrel_1(A,B,C),E)
                       => r1_tarski(D,E) ) ) ) ) ) ) ) )).

fof(t21_eqrel_1,axiom,(
    $true )).

fof(t22_eqrel_1,axiom,(
    ! [A,B] :
      ( ( v3_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => k5_eqrel_1(A,B,B) = B ) )).

fof(t23_eqrel_1,axiom,(
    ! [A,B] :
      ( ( v3_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] :
          ( ( v3_relat_2(C)
            & v8_relat_2(C)
            & v1_partfun1(C,A,A)
            & m2_relset_1(C,A,A) )
         => k5_eqrel_1(A,B,C) = k5_eqrel_1(A,C,B) ) ) )).

fof(t24_eqrel_1,axiom,(
    ! [A,B] :
      ( ( v3_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] :
          ( ( v3_relat_2(C)
            & v8_relat_2(C)
            & v1_partfun1(C,A,A)
            & m2_relset_1(C,A,A) )
         => k4_eqrel_1(A,B,k5_eqrel_1(A,B,C)) = B ) ) )).

fof(t25_eqrel_1,axiom,(
    ! [A,B] :
      ( ( v3_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] :
          ( ( v3_relat_2(C)
            & v8_relat_2(C)
            & v1_partfun1(C,A,A)
            & m2_relset_1(C,A,A) )
         => k5_eqrel_1(A,B,k4_eqrel_1(A,B,C)) = B ) ) )).

fof(d4_eqrel_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] : k6_eqrel_1(A,B,C) = k10_relset_1(A,A,B,k1_tarski(C)) ) )).

fof(t26_eqrel_1,axiom,(
    $true )).

fof(t27_eqrel_1,axiom,(
    ! [A,B,C,D] :
      ( ( v1_relat_2(D)
        & v3_relat_2(D)
        & v1_partfun1(D,A,A)
        & m2_relset_1(D,A,A) )
     => ( r2_hidden(B,k6_eqrel_1(A,D,C))
      <=> r2_hidden(k4_tarski(B,C),D) ) ) )).

fof(t28_eqrel_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] :
          ( r2_hidden(C,A)
         => r2_hidden(C,k6_eqrel_1(A,B,C)) ) ) )).

fof(t29_eqrel_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] :
          ~ ( r2_hidden(C,A)
            & ! [D] : ~ r2_hidden(C,k6_eqrel_1(A,B,D)) ) ) )).

fof(t30_eqrel_1,axiom,(
    ! [A,B,C,D,E] :
      ( ( v1_relat_2(E)
        & v3_relat_2(E)
        & v8_relat_2(E)
        & v1_partfun1(E,A,A)
        & m2_relset_1(E,A,A) )
     => ( ( r2_hidden(B,k6_eqrel_1(A,E,C))
          & r2_hidden(D,k6_eqrel_1(A,E,C)) )
       => r2_hidden(k4_tarski(B,D),E) ) ) )).

fof(t31_eqrel_1,axiom,(
    ! [A,B,C] :
      ( ( v3_relat_2(C)
        & v8_relat_2(C)
        & v1_partfun1(C,A,A)
        & m2_relset_1(C,A,A) )
     => ! [D] :
          ( r2_hidden(D,A)
         => ( r2_hidden(B,k6_eqrel_1(A,C,D))
          <=> k6_eqrel_1(A,C,D) = k6_eqrel_1(A,C,B) ) ) ) )).

fof(t32_eqrel_1,axiom,(
    ! [A,B] :
      ( ( v3_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C,D] :
          ~ ( r2_hidden(C,A)
            & r2_hidden(D,A)
            & k6_eqrel_1(A,B,C) != k6_eqrel_1(A,B,D)
            & ~ r1_xboole_0(k6_eqrel_1(A,B,C),k6_eqrel_1(A,B,D)) ) ) )).

fof(t33_eqrel_1,axiom,(
    ! [A,B] :
      ( r2_hidden(B,A)
     => k6_eqrel_1(A,k6_partfun1(A),B) = k1_tarski(B) ) )).

fof(t34_eqrel_1,axiom,(
    ! [A,B] :
      ( r2_hidden(B,A)
     => k6_eqrel_1(A,k1_eqrel_1(A),B) = A ) )).

fof(t35_eqrel_1,axiom,(
    ! [A,B] :
      ( ( v3_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ( ? [C] : k6_eqrel_1(A,B,C) = A
       => B = k1_eqrel_1(A) ) ) )).

fof(t36_eqrel_1,axiom,(
    ! [A,B,C,D] :
      ( ( v3_relat_2(D)
        & v8_relat_2(D)
        & v1_partfun1(D,B,B)
        & m2_relset_1(D,B,B) )
     => ! [E] :
          ( ( v3_relat_2(E)
            & v8_relat_2(E)
            & v1_partfun1(E,B,B)
            & m2_relset_1(E,B,B) )
         => ( r2_hidden(A,B)
           => ( r2_hidden(k4_tarski(A,C),k5_eqrel_1(B,D,E))
            <=> ? [F] :
                  ( v1_relat_1(F)
                  & v1_funct_1(F)
                  & v1_finseq_1(F)
                  & r1_xreal_0(np__1,k3_finseq_1(F))
                  & A = k1_funct_1(F,np__1)
                  & C = k1_funct_1(F,k3_finseq_1(F))
                  & ! [G] :
                      ( m2_subset_1(G,k1_numbers,k5_numbers)
                     => ( r1_xreal_0(np__1,G)
                       => ( r1_xreal_0(k3_finseq_1(F),G)
                          | r2_hidden(k4_tarski(k1_funct_1(F,G),k1_funct_1(F,k1_nat_1(G,np__1))),k3_eqrel_1(B,D,E)) ) ) ) ) ) ) ) ) )).

fof(t37_eqrel_1,axiom,(
    ! [A,B] :
      ( ( v3_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] :
          ( ( v3_relat_2(C)
            & v8_relat_2(C)
            & v1_partfun1(C,A,A)
            & m2_relset_1(C,A,A) )
         => ! [D] :
              ( ( v3_relat_2(D)
                & v8_relat_2(D)
                & v1_partfun1(D,A,A)
                & m2_relset_1(D,A,A) )
             => ( D = k3_eqrel_1(A,B,C)
               => ! [E] :
                    ~ ( r2_hidden(E,A)
                      & k6_eqrel_1(A,D,E) != k6_eqrel_1(A,B,E)
                      & k6_eqrel_1(A,D,E) != k6_eqrel_1(A,C,E) ) ) ) ) ) )).

fof(t38_eqrel_1,axiom,(
    ! [A,B] :
      ( ( v3_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] :
          ( ( v3_relat_2(C)
            & v8_relat_2(C)
            & v1_partfun1(C,A,A)
            & m2_relset_1(C,A,A) )
         => ~ ( k3_eqrel_1(A,B,C) = k1_eqrel_1(A)
              & B != k1_eqrel_1(A)
              & C != k1_eqrel_1(A) ) ) ) )).

fof(d5_eqrel_1,axiom,(
    ! [A,B] :
      ( ( v3_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
         => ( C = k7_eqrel_1(A,B)
          <=> ! [D] :
                ( m1_subset_1(D,k1_zfmisc_1(A))
               => ( r2_hidden(D,C)
                <=> ? [E] :
                      ( r2_hidden(E,A)
                      & D = k6_eqrel_1(A,B,E) ) ) ) ) ) ) )).

fof(t39_eqrel_1,axiom,(
    $true )).

fof(t40_eqrel_1,axiom,(
    ! [A,B] :
      ( ( v3_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ( A = k1_xboole_0
       => k7_eqrel_1(A,B) = k1_xboole_0 ) ) )).

fof(d6_eqrel_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( ( A != k1_xboole_0
         => ( m1_eqrel_1(B,A)
          <=> ( k5_setfam_1(A,B) = A
              & ! [C] :
                  ( m1_subset_1(C,k1_zfmisc_1(A))
                 => ( r2_hidden(C,B)
                   => ( C != k1_xboole_0
                      & ! [D] :
                          ( m1_subset_1(D,k1_zfmisc_1(A))
                         => ~ ( r2_hidden(D,B)
                              & C != D
                              & ~ r1_xboole_0(C,D) ) ) ) ) ) ) ) )
        & ( A = k1_xboole_0
         => ( m1_eqrel_1(B,A)
          <=> B = k1_xboole_0 ) ) ) ) )).

fof(t41_eqrel_1,axiom,(
    $true )).

fof(t42_eqrel_1,axiom,(
    ! [A,B] :
      ( ( v3_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => m1_eqrel_1(k7_eqrel_1(A,B),A) ) )).

fof(t43_eqrel_1,axiom,(
    ! [A,B] :
      ( m1_eqrel_1(B,A)
     => ? [C] :
          ( v3_relat_2(C)
          & v8_relat_2(C)
          & v1_partfun1(C,A,A)
          & m2_relset_1(C,A,A)
          & B = k7_eqrel_1(A,C) ) ) )).

fof(t44_eqrel_1,axiom,(
    ! [A,B,C] :
      ( ( v3_relat_2(C)
        & v8_relat_2(C)
        & v1_partfun1(C,A,A)
        & m2_relset_1(C,A,A) )
     => ! [D] :
          ( r2_hidden(D,A)
         => ( r2_hidden(k4_tarski(D,B),C)
          <=> k6_eqrel_1(A,C,D) = k6_eqrel_1(A,C,B) ) ) ) )).

fof(t45_eqrel_1,axiom,(
    ! [A,B,C] :
      ( ( v3_relat_2(C)
        & v8_relat_2(C)
        & v1_partfun1(C,B,B)
        & m2_relset_1(C,B,B) )
     => ~ ( r2_hidden(A,k7_eqrel_1(B,C))
          & ! [D] :
              ( m1_subset_1(D,B)
             => A != k6_eqrel_1(B,C,D) ) ) ) )).

fof(s1_eqrel_1,axiom,
    ( ( ! [A] :
          ( r2_hidden(A,f1_s1_eqrel_1)
         => p1_s1_eqrel_1(A,A) )
      & ! [A,B] :
          ( p1_s1_eqrel_1(A,B)
         => p1_s1_eqrel_1(B,A) )
      & ! [A,B,C] :
          ( ( p1_s1_eqrel_1(A,B)
            & p1_s1_eqrel_1(B,C) )
         => p1_s1_eqrel_1(A,C) ) )
   => ? [A] :
        ( v3_relat_2(A)
        & v8_relat_2(A)
        & v1_partfun1(A,f1_s1_eqrel_1,f1_s1_eqrel_1)
        & m2_relset_1(A,f1_s1_eqrel_1,f1_s1_eqrel_1)
        & ! [B,C] :
            ( r2_hidden(k4_tarski(B,C),A)
          <=> ( r2_hidden(B,f1_s1_eqrel_1)
              & r2_hidden(C,f1_s1_eqrel_1)
              & p1_s1_eqrel_1(B,C) ) ) ) )).

fof(dt_m1_eqrel_1,axiom,(
    ! [A,B] :
      ( m1_eqrel_1(B,A)
     => m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) )).

fof(existence_m1_eqrel_1,axiom,(
    ! [A] :
    ? [B] : m1_eqrel_1(B,A) )).

fof(dt_k1_eqrel_1,axiom,(
    ! [A] : m2_relset_1(k1_eqrel_1(A),A,A) )).

fof(dt_k2_eqrel_1,axiom,(
    ! [A,B,C] :
      ( ( m1_relset_1(B,A,A)
        & m1_relset_1(C,A,A) )
     => m2_relset_1(k2_eqrel_1(A,B,C),A,A) ) )).

fof(commutativity_k2_eqrel_1,axiom,(
    ! [A,B,C] :
      ( ( m1_relset_1(B,A,A)
        & m1_relset_1(C,A,A) )
     => k2_eqrel_1(A,B,C) = k2_eqrel_1(A,C,B) ) )).

fof(idempotence_k2_eqrel_1,axiom,(
    ! [A,B,C] :
      ( ( m1_relset_1(B,A,A)
        & m1_relset_1(C,A,A) )
     => k2_eqrel_1(A,B,B) = B ) )).

fof(redefinition_k2_eqrel_1,axiom,(
    ! [A,B,C] :
      ( ( m1_relset_1(B,A,A)
        & m1_relset_1(C,A,A) )
     => k2_eqrel_1(A,B,C) = k3_xboole_0(B,C) ) )).

fof(dt_k3_eqrel_1,axiom,(
    ! [A,B,C] :
      ( ( m1_relset_1(B,A,A)
        & m1_relset_1(C,A,A) )
     => m2_relset_1(k3_eqrel_1(A,B,C),A,A) ) )).

fof(commutativity_k3_eqrel_1,axiom,(
    ! [A,B,C] :
      ( ( m1_relset_1(B,A,A)
        & m1_relset_1(C,A,A) )
     => k3_eqrel_1(A,B,C) = k3_eqrel_1(A,C,B) ) )).

fof(idempotence_k3_eqrel_1,axiom,(
    ! [A,B,C] :
      ( ( m1_relset_1(B,A,A)
        & m1_relset_1(C,A,A) )
     => k3_eqrel_1(A,B,B) = B ) )).

fof(redefinition_k3_eqrel_1,axiom,(
    ! [A,B,C] :
      ( ( m1_relset_1(B,A,A)
        & m1_relset_1(C,A,A) )
     => k3_eqrel_1(A,B,C) = k2_xboole_0(B,C) ) )).

fof(dt_k4_eqrel_1,axiom,(
    ! [A,B,C] :
      ( ( v3_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m1_relset_1(B,A,A)
        & v3_relat_2(C)
        & v8_relat_2(C)
        & v1_partfun1(C,A,A)
        & m1_relset_1(C,A,A) )
     => ( v3_relat_2(k4_eqrel_1(A,B,C))
        & v8_relat_2(k4_eqrel_1(A,B,C))
        & v1_partfun1(k4_eqrel_1(A,B,C),A,A)
        & m2_relset_1(k4_eqrel_1(A,B,C),A,A) ) ) )).

fof(commutativity_k4_eqrel_1,axiom,(
    ! [A,B,C] :
      ( ( v3_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m1_relset_1(B,A,A)
        & v3_relat_2(C)
        & v8_relat_2(C)
        & v1_partfun1(C,A,A)
        & m1_relset_1(C,A,A) )
     => k4_eqrel_1(A,B,C) = k4_eqrel_1(A,C,B) ) )).

fof(idempotence_k4_eqrel_1,axiom,(
    ! [A,B,C] :
      ( ( v3_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m1_relset_1(B,A,A)
        & v3_relat_2(C)
        & v8_relat_2(C)
        & v1_partfun1(C,A,A)
        & m1_relset_1(C,A,A) )
     => k4_eqrel_1(A,B,B) = B ) )).

fof(redefinition_k4_eqrel_1,axiom,(
    ! [A,B,C] :
      ( ( v3_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m1_relset_1(B,A,A)
        & v3_relat_2(C)
        & v8_relat_2(C)
        & v1_partfun1(C,A,A)
        & m1_relset_1(C,A,A) )
     => k4_eqrel_1(A,B,C) = k3_xboole_0(B,C) ) )).

fof(dt_k5_eqrel_1,axiom,(
    ! [A,B,C] :
      ( ( v3_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m1_relset_1(B,A,A)
        & v3_relat_2(C)
        & v8_relat_2(C)
        & v1_partfun1(C,A,A)
        & m1_relset_1(C,A,A) )
     => ( v3_relat_2(k5_eqrel_1(A,B,C))
        & v8_relat_2(k5_eqrel_1(A,B,C))
        & v1_partfun1(k5_eqrel_1(A,B,C),A,A)
        & m2_relset_1(k5_eqrel_1(A,B,C),A,A) ) ) )).

fof(commutativity_k5_eqrel_1,axiom,(
    ! [A,B,C] :
      ( ( v3_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m1_relset_1(B,A,A)
        & v3_relat_2(C)
        & v8_relat_2(C)
        & v1_partfun1(C,A,A)
        & m1_relset_1(C,A,A) )
     => k5_eqrel_1(A,B,C) = k5_eqrel_1(A,C,B) ) )).

fof(idempotence_k5_eqrel_1,axiom,(
    ! [A,B,C] :
      ( ( v3_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m1_relset_1(B,A,A)
        & v3_relat_2(C)
        & v8_relat_2(C)
        & v1_partfun1(C,A,A)
        & m1_relset_1(C,A,A) )
     => k5_eqrel_1(A,B,B) = B ) )).

fof(dt_k6_eqrel_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m1_relset_1(B,A,A) )
     => m1_subset_1(k6_eqrel_1(A,B,C),k1_zfmisc_1(A)) ) )).

fof(dt_k7_eqrel_1,axiom,(
    ! [A,B] :
      ( ( v3_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m1_relset_1(B,A,A) )
     => m1_subset_1(k7_eqrel_1(A,B),k1_zfmisc_1(k1_zfmisc_1(A))) ) )).

fof(dt_k8_eqrel_1,axiom,(
    ! [A,B] :
      ( ( v3_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m1_relset_1(B,A,A) )
     => m1_eqrel_1(k8_eqrel_1(A,B),A) ) )).

fof(redefinition_k8_eqrel_1,axiom,(
    ! [A,B] :
      ( ( v3_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m1_relset_1(B,A,A) )
     => k8_eqrel_1(A,B) = k7_eqrel_1(A,B) ) )).
%------------------------------------------------------------------------------