SET007 Axioms: SET007+7.ax


%------------------------------------------------------------------------------
% File     : SET007+7 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Properties of Subsets
% 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    : subset_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  110 (  30 unit)
%            Number of atoms       :  368 (  48 equality)
%            Maximal formula depth :   20 (   6 average)
%            Number of connectives :  316 (  58 ~  ;   1  |;  47  &)
%                                         (  25 <=>; 185 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   14 (   1 propositional; 0-3 arity)
%            Number of functors    :   31 (   7 constant; 0-8 arity)
%            Number of variables   :  265 (   0 singleton; 259 !;   6 ?)
%            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_subset_1,axiom,(
    ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) )).

fof(fc2_subset_1,axiom,(
    ! [A] : ~ v1_xboole_0(k1_tarski(A)) )).

fof(fc3_subset_1,axiom,(
    ! [A,B] : ~ v1_xboole_0(k2_tarski(A,B)) )).

fof(rc1_subset_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
          & ~ v1_xboole_0(B) ) ) )).

fof(fc4_subset_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B) )
     => ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) )).

fof(fc5_subset_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C) )
     => ~ v1_xboole_0(k3_zfmisc_1(A,B,C)) ) )).

fof(fc6_subset_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & ~ v1_xboole_0(D) )
     => ~ v1_xboole_0(k4_zfmisc_1(A,B,C,D)) ) )).

fof(rc2_subset_1,axiom,(
    ! [A] :
    ? [B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
      & v1_xboole_0(B) ) )).

fof(d1_subset_1,axiom,(
    $true )).

fof(d2_subset_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
       => ( m1_subset_1(B,A)
        <=> r2_hidden(B,A) ) )
      & ( v1_xboole_0(A)
       => ( m1_subset_1(B,A)
        <=> v1_xboole_0(B) ) ) ) )).

fof(d3_subset_1,axiom,(
    ! [A] : k1_subset_1(A) = k1_xboole_0 )).

fof(d4_subset_1,axiom,(
    ! [A] : k2_subset_1(A) = A )).

fof(t1_subset_1,axiom,(
    $true )).

fof(t2_subset_1,axiom,(
    $true )).

fof(t3_subset_1,axiom,(
    $true )).

fof(t4_subset_1,axiom,(
    ! [A] : m1_subset_1(k1_xboole_0,k1_zfmisc_1(A)) )).

fof(t5_subset_1,axiom,(
    $true )).

fof(t6_subset_1,axiom,(
    $true )).

fof(t7_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ( ! [D] :
                ( m1_subset_1(D,A)
               => ( r2_hidden(D,B)
                 => r2_hidden(D,C) ) )
           => r1_tarski(B,C) ) ) ) )).

fof(t8_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ( ! [D] :
                ( m1_subset_1(D,A)
               => ( r2_hidden(D,B)
                <=> r2_hidden(D,C) ) )
           => B = C ) ) ) )).

fof(t9_subset_1,axiom,(
    $true )).

fof(t10_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ~ ( B != k1_xboole_0
          & ! [C] :
              ( m1_subset_1(C,A)
             => ~ r2_hidden(C,B) ) ) ) )).

fof(d5_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => k3_subset_1(A,B) = k4_xboole_0(A,B) ) )).

fof(t11_subset_1,axiom,(
    $true )).

fof(t12_subset_1,axiom,(
    $true )).

fof(t13_subset_1,axiom,(
    $true )).

fof(t14_subset_1,axiom,(
    $true )).

fof(t15_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(A))
             => ( ! [E] :
                    ( m1_subset_1(E,A)
                   => ( r2_hidden(E,B)
                    <=> ( r2_hidden(E,C)
                        | r2_hidden(E,D) ) ) )
               => B = k4_subset_1(A,C,D) ) ) ) ) )).

fof(t16_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(A))
             => ( ! [E] :
                    ( m1_subset_1(E,A)
                   => ( r2_hidden(E,B)
                    <=> ( r2_hidden(E,C)
                        & r2_hidden(E,D) ) ) )
               => B = k5_subset_1(A,C,D) ) ) ) ) )).

fof(t17_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(A))
             => ( ! [E] :
                    ( m1_subset_1(E,A)
                   => ( r2_hidden(E,B)
                    <=> ( r2_hidden(E,C)
                        & ~ r2_hidden(E,D) ) ) )
               => B = k6_subset_1(A,C,D) ) ) ) ) )).

fof(t18_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(A))
             => ( ! [E] :
                    ( m1_subset_1(E,A)
                   => ( r2_hidden(E,B)
                    <=> ~ ( r2_hidden(E,C)
                        <=> r2_hidden(E,D) ) ) )
               => B = k7_subset_1(A,C,D) ) ) ) ) )).

fof(t19_subset_1,axiom,(
    $true )).

fof(t20_subset_1,axiom,(
    $true )).

fof(t21_subset_1,axiom,(
    ! [A] : k1_subset_1(A) = k3_subset_1(A,k2_subset_1(A)) )).

fof(t22_subset_1,axiom,(
    ! [A] : k2_subset_1(A) = k3_subset_1(A,k1_subset_1(A)) )).

fof(t23_subset_1,axiom,(
    $true )).

fof(t24_subset_1,axiom,(
    $true )).

fof(t25_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => k4_subset_1(A,B,k3_subset_1(A,B)) = k2_subset_1(A) ) )).

fof(t26_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => r1_xboole_0(B,k3_subset_1(A,B)) ) )).

fof(t27_subset_1,axiom,(
    $true )).

fof(t28_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => k4_subset_1(A,B,k2_subset_1(A)) = k2_subset_1(A) ) )).

fof(t29_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => k3_subset_1(A,k4_subset_1(A,B,C)) = k5_subset_1(A,k3_subset_1(A,B),k3_subset_1(A,C)) ) ) )).

fof(t30_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => k3_subset_1(A,k5_subset_1(A,B,C)) = k4_subset_1(A,k3_subset_1(A,B),k3_subset_1(A,C)) ) ) )).

fof(t31_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ( r1_tarski(B,C)
          <=> r1_tarski(k3_subset_1(A,C),k3_subset_1(A,B)) ) ) ) )).

fof(t32_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => k6_subset_1(A,B,C) = k5_subset_1(A,B,k3_subset_1(A,C)) ) ) )).

fof(t33_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => k3_subset_1(A,k6_subset_1(A,B,C)) = k4_subset_1(A,k3_subset_1(A,B),C) ) ) )).

fof(t34_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => k3_subset_1(A,k7_subset_1(A,B,C)) = k4_subset_1(A,k5_subset_1(A,B,C),k5_subset_1(A,k3_subset_1(A,B),k3_subset_1(A,C))) ) ) )).

fof(t35_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ( r1_tarski(B,k3_subset_1(A,C))
           => r1_tarski(C,k3_subset_1(A,B)) ) ) ) )).

fof(t36_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ( r1_tarski(k3_subset_1(A,B),C)
           => r1_tarski(k3_subset_1(A,C),B) ) ) ) )).

fof(t37_subset_1,axiom,(
    $true )).

fof(t38_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ( r1_tarski(B,k3_subset_1(A,B))
      <=> B = k1_subset_1(A) ) ) )).

fof(t39_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ( r1_tarski(k3_subset_1(A,B),B)
      <=> B = k2_subset_1(A) ) ) )).

fof(t40_subset_1,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => ( ( r1_tarski(B,C)
          & r1_tarski(B,k3_subset_1(A,C)) )
       => B = k1_xboole_0 ) ) )).

fof(t41_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => r1_tarski(k3_subset_1(A,k4_subset_1(A,B,C)),k3_subset_1(A,B)) ) ) )).

fof(t42_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => r1_tarski(k3_subset_1(A,B),k3_subset_1(A,k5_subset_1(A,B,C))) ) ) )).

fof(t43_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ( r1_xboole_0(B,C)
          <=> r1_tarski(B,k3_subset_1(A,C)) ) ) ) )).

fof(t44_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ( r1_xboole_0(B,k3_subset_1(A,C))
          <=> r1_tarski(B,C) ) ) ) )).

fof(t45_subset_1,axiom,(
    $true )).

fof(t46_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ( ( r1_xboole_0(B,C)
              & r1_xboole_0(k3_subset_1(A,B),k3_subset_1(A,C)) )
           => B = k3_subset_1(A,C) ) ) ) )).

fof(t47_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(A))
             => ( ( r1_tarski(B,C)
                  & r1_xboole_0(D,C) )
               => r1_tarski(B,k3_subset_1(A,D)) ) ) ) ) )).

fof(t48_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ( ! [D] :
                ( m1_subset_1(D,B)
               => r2_hidden(D,C) )
           => r1_tarski(B,C) ) ) ) )).

fof(t49_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ( ! [C] :
            ( m1_subset_1(C,A)
           => r2_hidden(C,B) )
       => A = B ) ) )).

fof(t50_subset_1,axiom,(
    ! [A] :
      ( A != k1_xboole_0
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m1_subset_1(C,A)
             => ( ~ r2_hidden(C,B)
               => r2_hidden(C,k3_subset_1(A,B)) ) ) ) ) )).

fof(t51_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ( ! [D] :
                ( m1_subset_1(D,A)
               => ( r2_hidden(D,B)
                <=> ~ r2_hidden(D,C) ) )
           => B = k3_subset_1(A,C) ) ) ) )).

fof(t52_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ( ! [D] :
                ( m1_subset_1(D,A)
               => ( ~ r2_hidden(D,B)
                <=> r2_hidden(D,C) ) )
           => B = k3_subset_1(A,C) ) ) ) )).

fof(t53_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ( ! [D] :
                ( m1_subset_1(D,A)
               => ~ ( r2_hidden(D,B)
                  <=> r2_hidden(D,C) ) )
           => B = k3_subset_1(A,C) ) ) ) )).

fof(t54_subset_1,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => ~ ( r2_hidden(B,k3_subset_1(A,C))
          & r2_hidden(B,C) ) ) )).

fof(t55_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,A)
     => ( A != k1_xboole_0
       => m1_subset_1(k1_tarski(B),k1_zfmisc_1(A)) ) ) )).

fof(t56_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,A)
     => ! [C] :
          ( m1_subset_1(C,A)
         => ( A != k1_xboole_0
           => m1_subset_1(k2_tarski(B,C),k1_zfmisc_1(A)) ) ) ) )).

fof(t57_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,A)
     => ! [C] :
          ( m1_subset_1(C,A)
         => ! [D] :
              ( m1_subset_1(D,A)
             => ( A != k1_xboole_0
               => m1_subset_1(k1_enumset1(B,C,D),k1_zfmisc_1(A)) ) ) ) ) )).

fof(t58_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,A)
     => ! [C] :
          ( m1_subset_1(C,A)
         => ! [D] :
              ( m1_subset_1(D,A)
             => ! [E] :
                  ( m1_subset_1(E,A)
                 => ( A != k1_xboole_0
                   => m1_subset_1(k2_enumset1(B,C,D,E),k1_zfmisc_1(A)) ) ) ) ) ) )).

fof(t59_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,A)
     => ! [C] :
          ( m1_subset_1(C,A)
         => ! [D] :
              ( m1_subset_1(D,A)
             => ! [E] :
                  ( m1_subset_1(E,A)
                 => ! [F] :
                      ( m1_subset_1(F,A)
                     => ( A != k1_xboole_0
                       => m1_subset_1(k3_enumset1(B,C,D,E,F),k1_zfmisc_1(A)) ) ) ) ) ) ) )).

fof(t60_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,A)
     => ! [C] :
          ( m1_subset_1(C,A)
         => ! [D] :
              ( m1_subset_1(D,A)
             => ! [E] :
                  ( m1_subset_1(E,A)
                 => ! [F] :
                      ( m1_subset_1(F,A)
                     => ! [G] :
                          ( m1_subset_1(G,A)
                         => ( A != k1_xboole_0
                           => m1_subset_1(k4_enumset1(B,C,D,E,F,G),k1_zfmisc_1(A)) ) ) ) ) ) ) ) )).

fof(t61_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,A)
     => ! [C] :
          ( m1_subset_1(C,A)
         => ! [D] :
              ( m1_subset_1(D,A)
             => ! [E] :
                  ( m1_subset_1(E,A)
                 => ! [F] :
                      ( m1_subset_1(F,A)
                     => ! [G] :
                          ( m1_subset_1(G,A)
                         => ! [H] :
                              ( m1_subset_1(H,A)
                             => ( A != k1_xboole_0
                               => m1_subset_1(k5_enumset1(B,C,D,E,F,G,H),k1_zfmisc_1(A)) ) ) ) ) ) ) ) ) )).

fof(t62_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,A)
     => ! [C] :
          ( m1_subset_1(C,A)
         => ! [D] :
              ( m1_subset_1(D,A)
             => ! [E] :
                  ( m1_subset_1(E,A)
                 => ! [F] :
                      ( m1_subset_1(F,A)
                     => ! [G] :
                          ( m1_subset_1(G,A)
                         => ! [H] :
                              ( m1_subset_1(H,A)
                             => ! [I] :
                                  ( m1_subset_1(I,A)
                                 => ( A != k1_xboole_0
                                   => m1_subset_1(k6_enumset1(B,C,D,E,F,G,H,I),k1_zfmisc_1(A)) ) ) ) ) ) ) ) ) ) )).

fof(t63_subset_1,axiom,(
    ! [A,B] :
      ( r2_hidden(A,B)
     => m1_subset_1(k1_tarski(A),k1_zfmisc_1(B)) ) )).

fof(d6_subset_1,axiom,(
    ! [A] :
      ( ~ $true
     => ! [B] :
          ( m1_subset_1(B,A)
         => B = k8_subset_1(A) ) ) )).

fof(s1_subset_1,axiom,(
    ? [A] :
      ( m1_subset_1(A,k1_zfmisc_1(f1_s1_subset_1))
      & ! [B] :
          ( r2_hidden(B,A)
        <=> ( r2_hidden(B,f1_s1_subset_1)
            & p1_s1_subset_1(B) ) ) ) )).

fof(s2_subset_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(f1_s2_subset_1))
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(f1_s2_subset_1))
         => ( ( ! [C] :
                  ( m1_subset_1(C,f1_s2_subset_1)
                 => ( r2_hidden(C,A)
                  <=> p1_s2_subset_1(C) ) )
              & ! [C] :
                  ( m1_subset_1(C,f1_s2_subset_1)
                 => ( r2_hidden(C,B)
                  <=> p1_s2_subset_1(C) ) ) )
           => A = B ) ) ) )).

fof(s3_subset_1,axiom,(
    ? [A] :
      ( m1_subset_1(A,k1_zfmisc_1(f1_s3_subset_1))
      & ! [B] :
          ( m1_subset_1(B,f1_s3_subset_1)
         => ( r2_hidden(B,A)
          <=> p1_s3_subset_1(B) ) ) ) )).

fof(s4_subset_1,axiom,
    ( ( ! [A] :
          ( m1_subset_1(A,f1_s4_subset_1)
         => ( r2_hidden(A,f2_s4_subset_1)
          <=> p1_s4_subset_1(A) ) )
      & ! [A] :
          ( m1_subset_1(A,f1_s4_subset_1)
         => ( r2_hidden(A,f3_s4_subset_1)
          <=> p1_s4_subset_1(A) ) ) )
   => f2_s4_subset_1 = f3_s4_subset_1 )).

fof(dt_m1_subset_1,axiom,(
    $true )).

fof(existence_m1_subset_1,axiom,(
    ! [A] :
    ? [B] : m1_subset_1(B,A) )).

fof(dt_m2_subset_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ! [C] :
          ( m2_subset_1(C,A,B)
         => m1_subset_1(C,A) ) ) )).

fof(existence_m2_subset_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ? [C] : m2_subset_1(C,A,B) ) )).

fof(redefinition_m2_subset_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ! [C] :
          ( m2_subset_1(C,A,B)
        <=> m1_subset_1(C,B) ) ) )).

fof(symmetry_r1_subset_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B) )
     => ( r1_subset_1(A,B)
       => r1_subset_1(B,A) ) ) )).

fof(irreflexivity_r1_subset_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B) )
     => ~ r1_subset_1(A,A) ) )).

fof(redefinition_r1_subset_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B) )
     => ( r1_subset_1(A,B)
      <=> r1_xboole_0(A,B) ) ) )).

fof(symmetry_r2_subset_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B) )
     => ( r2_subset_1(A,B)
       => r2_subset_1(B,A) ) ) )).

fof(irreflexivity_r2_subset_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B) )
     => ~ r2_subset_1(A,A) ) )).

fof(redefinition_r2_subset_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B) )
     => ( r2_subset_1(A,B)
      <=> r1_xboole_0(A,B) ) ) )).

fof(dt_k1_subset_1,axiom,(
    ! [A] :
      ( v1_xboole_0(k1_subset_1(A))
      & m1_subset_1(k1_subset_1(A),k1_zfmisc_1(A)) ) )).

fof(dt_k2_subset_1,axiom,(
    ! [A] : m1_subset_1(k2_subset_1(A),k1_zfmisc_1(A)) )).

fof(dt_k3_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => m1_subset_1(k3_subset_1(A,B),k1_zfmisc_1(A)) ) )).

fof(involutiveness_k3_subset_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => k3_subset_1(A,k3_subset_1(A,B)) = B ) )).

fof(dt_k4_subset_1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => m1_subset_1(k4_subset_1(A,B,C),k1_zfmisc_1(A)) ) )).

fof(commutativity_k4_subset_1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => k4_subset_1(A,B,C) = k4_subset_1(A,C,B) ) )).

fof(idempotence_k4_subset_1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => k4_subset_1(A,B,B) = B ) )).

fof(redefinition_k4_subset_1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => k4_subset_1(A,B,C) = k2_xboole_0(B,C) ) )).

fof(dt_k5_subset_1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => m1_subset_1(k5_subset_1(A,B,C),k1_zfmisc_1(A)) ) )).

fof(commutativity_k5_subset_1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => k5_subset_1(A,B,C) = k5_subset_1(A,C,B) ) )).

fof(idempotence_k5_subset_1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => k5_subset_1(A,B,B) = B ) )).

fof(redefinition_k5_subset_1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => k5_subset_1(A,B,C) = k3_xboole_0(B,C) ) )).

fof(dt_k6_subset_1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => m1_subset_1(k6_subset_1(A,B,C),k1_zfmisc_1(A)) ) )).

fof(redefinition_k6_subset_1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => k6_subset_1(A,B,C) = k4_xboole_0(B,C) ) )).

fof(dt_k7_subset_1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => m1_subset_1(k7_subset_1(A,B,C),k1_zfmisc_1(A)) ) )).

fof(commutativity_k7_subset_1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => k7_subset_1(A,B,C) = k7_subset_1(A,C,B) ) )).

fof(redefinition_k7_subset_1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => k7_subset_1(A,B,C) = k5_xboole_0(B,C) ) )).

fof(dt_k8_subset_1,axiom,(
    ! [A] : m1_subset_1(k8_subset_1(A),A) )).
%------------------------------------------------------------------------------