SET007 Axioms: SET007+377.ax


%------------------------------------------------------------------------------
% File     : SET007+377 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Representation Theorem for Boolean Algebras
% 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    : lopclset [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   84 (   6 unit)
%            Number of atoms       :  548 (  50 equality)
%            Maximal formula depth :   19 (   8 average)
%            Number of connectives :  603 ( 139 ~  ;   1  |; 323  &)
%                                         (  13 <=>; 127 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   40 (   1 propositional; 0-3 arity)
%            Number of functors    :   46 (   1 constant; 0-6 arity)
%            Number of variables   :  156 (   0 singleton; 146 !;  10 ?)
%            Maximal term depth    :    5 (   1 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(fc1_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ~ v1_xboole_0(k1_lopclset(A)) ) )).

fof(rc1_lopclset,axiom,(
    ? [A] :
      ( l3_lattices(A)
      & ~ v3_struct_0(A)
      & v4_lattices(A)
      & v5_lattices(A)
      & v6_lattices(A)
      & v7_lattices(A)
      & v8_lattices(A)
      & v9_lattices(A)
      & v10_lattices(A)
      & v11_lattices(A)
      & v12_lattices(A)
      & v13_lattices(A)
      & v14_lattices(A)
      & v15_lattices(A)
      & v16_lattices(A)
      & v17_lattices(A)
      & ~ v3_realset2(A) ) )).

fof(fc2_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => ~ v1_xboole_0(k7_lopclset(A)) ) )).

fof(fc3_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => ~ v1_xboole_0(k10_lopclset(A)) ) )).

fof(fc4_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => ( ~ v3_struct_0(k11_lopclset(A))
        & v1_pre_topc(k11_lopclset(A))
        & v2_pre_topc(k11_lopclset(A)) ) ) )).

fof(rc2_lopclset,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ? [B] :
          ( m1_subset_1(B,k5_finsub_1(A))
          & ~ v1_xboole_0(B)
          & v1_finset_1(B) ) ) )).

fof(t1_lopclset,axiom,(
    $true )).

fof(t2_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( r2_hidden(B,k1_lopclset(A))
           => v3_pre_topc(B,A) ) ) ) )).

fof(t3_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( r2_hidden(B,k1_lopclset(A))
           => v4_pre_topc(B,A) ) ) ) )).

fof(t4_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( ( v3_pre_topc(B,A)
              & v4_pre_topc(B,A) )
           => r2_hidden(B,k1_lopclset(A)) ) ) ) )).

fof(d2_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(k1_lopclset(A),k1_lopclset(A)),k1_lopclset(A))
            & m2_relset_1(B,k2_zfmisc_1(k1_lopclset(A),k1_lopclset(A)),k1_lopclset(A)) )
         => ( B = k4_lopclset(A)
          <=> ! [C] :
                ( m2_subset_1(C,k1_zfmisc_1(u1_struct_0(A)),k1_lopclset(A))
               => ! [D] :
                    ( m2_subset_1(D,k1_zfmisc_1(u1_struct_0(A)),k1_lopclset(A))
                   => k2_binop_1(k1_lopclset(A),k1_lopclset(A),k1_lopclset(A),B,C,D) = k2_lopclset(A,C,D) ) ) ) ) ) )).

fof(d3_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(k1_lopclset(A),k1_lopclset(A)),k1_lopclset(A))
            & m2_relset_1(B,k2_zfmisc_1(k1_lopclset(A),k1_lopclset(A)),k1_lopclset(A)) )
         => ( B = k5_lopclset(A)
          <=> ! [C] :
                ( m2_subset_1(C,k1_zfmisc_1(u1_struct_0(A)),k1_lopclset(A))
               => ! [D] :
                    ( m2_subset_1(D,k1_zfmisc_1(u1_struct_0(A)),k1_lopclset(A))
                   => k2_binop_1(k1_lopclset(A),k1_lopclset(A),k1_lopclset(A),B,C,D) = k3_lopclset(A,C,D) ) ) ) ) ) )).

fof(t5_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(g3_lattices(k1_lopclset(A),k4_lopclset(A),k5_lopclset(A))))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(g3_lattices(k1_lopclset(A),k4_lopclset(A),k5_lopclset(A))))
             => ! [D] :
                  ( m2_subset_1(D,k1_zfmisc_1(u1_struct_0(A)),k1_lopclset(A))
                 => ! [E] :
                      ( m2_subset_1(E,k1_zfmisc_1(u1_struct_0(A)),k1_lopclset(A))
                     => ( ( B = D
                          & C = E )
                       => k1_lattices(g3_lattices(k1_lopclset(A),k4_lopclset(A),k5_lopclset(A)),B,C) = k2_lopclset(A,D,E) ) ) ) ) ) ) )).

fof(t6_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(g3_lattices(k1_lopclset(A),k4_lopclset(A),k5_lopclset(A))))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(g3_lattices(k1_lopclset(A),k4_lopclset(A),k5_lopclset(A))))
             => ! [D] :
                  ( m2_subset_1(D,k1_zfmisc_1(u1_struct_0(A)),k1_lopclset(A))
                 => ! [E] :
                      ( m2_subset_1(E,k1_zfmisc_1(u1_struct_0(A)),k1_lopclset(A))
                     => ( ( B = D
                          & C = E )
                       => k2_lattices(g3_lattices(k1_lopclset(A),k4_lopclset(A),k5_lopclset(A)),B,C) = k3_lopclset(A,D,E) ) ) ) ) ) ) )).

fof(t7_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => m2_subset_1(k1_pre_topc(A),k1_zfmisc_1(u1_struct_0(A)),k1_lopclset(A)) ) )).

fof(t8_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => m2_subset_1(k2_pre_topc(A),k1_zfmisc_1(u1_struct_0(A)),k1_lopclset(A)) ) )).

fof(t9_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_zfmisc_1(u1_struct_0(A)),k1_lopclset(A))
         => m2_subset_1(k3_subset_1(u1_struct_0(A),B),k1_zfmisc_1(u1_struct_0(A)),k1_lopclset(A)) ) ) )).

fof(t10_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( ~ v3_struct_0(g3_lattices(k1_lopclset(A),k4_lopclset(A),k5_lopclset(A)))
        & v10_lattices(g3_lattices(k1_lopclset(A),k4_lopclset(A),k5_lopclset(A)))
        & l3_lattices(g3_lattices(k1_lopclset(A),k4_lopclset(A),k5_lopclset(A))) ) ) )).

fof(d4_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => k6_lopclset(A) = g3_lattices(k1_lopclset(A),k4_lopclset(A),k5_lopclset(A)) ) )).

fof(t11_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k6_lopclset(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k6_lopclset(A)))
             => k3_lattices(k6_lopclset(A),B,C) = k2_xboole_0(B,C) ) ) ) )).

fof(t12_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k6_lopclset(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k6_lopclset(A)))
             => k4_lattices(k6_lopclset(A),B,C) = k3_xboole_0(B,C) ) ) ) )).

fof(t13_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => u1_struct_0(k6_lopclset(A)) = k1_lopclset(A) ) )).

fof(t14_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => v17_lattices(k6_lopclset(A)) ) )).

fof(t15_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => m1_subset_1(k2_pre_topc(A),u1_struct_0(k6_lopclset(A))) ) )).

fof(t16_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => m1_subset_1(k1_pre_topc(A),u1_struct_0(k6_lopclset(A))) ) )).

fof(t17_lopclset,axiom,(
    $true )).

fof(t18_lopclset,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & v10_lattices(B)
        & v17_lattices(B)
        & ~ v3_realset2(B)
        & l3_lattices(B) )
     => ( r2_hidden(A,k7_lopclset(B))
      <=> ? [C] :
            ( m1_filter_0(C,B)
            & C = A
            & v1_filter_0(C,B) ) ) ) )).

fof(t20_lopclset,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & v10_lattices(B)
        & v17_lattices(B)
        & ~ v3_realset2(B)
        & l3_lattices(B) )
     => ! [C] :
          ( m1_subset_1(C,u1_struct_0(B))
         => ( r2_hidden(A,k1_funct_1(k8_lopclset(B),C))
          <=> ? [D] :
                ( m1_filter_0(D,B)
                & D = A
                & v1_filter_0(D,B)
                & r2_hidden(C,D) ) ) ) ) )).

fof(t21_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_filter_0(C,A)
             => ( r2_hidden(C,k1_funct_1(k8_lopclset(A),B))
              <=> ( v1_filter_0(C,A)
                  & r2_hidden(B,C) ) ) ) ) ) )).

fof(t22_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_filter_0(D,A)
                 => ( v1_filter_0(D,A)
                   => ( r2_hidden(k3_lattices(A,B,C),D)
                    <=> ( r2_hidden(B,D)
                        | r2_hidden(C,D) ) ) ) ) ) ) ) )).

fof(t23_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k1_funct_1(k8_lopclset(A),k4_lattices(A,B,C)) = k3_xboole_0(k1_funct_1(k8_lopclset(A),B),k1_funct_1(k8_lopclset(A),C)) ) ) ) )).

fof(t24_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k1_funct_1(k8_lopclset(A),k3_lattices(A,B,C)) = k2_xboole_0(k1_funct_1(k8_lopclset(A),B),k1_funct_1(k8_lopclset(A),C)) ) ) ) )).

fof(d7_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => k10_lopclset(A) = k2_relat_1(k9_lopclset(A)) ) )).

fof(t25_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => r1_tarski(k10_lopclset(A),k1_zfmisc_1(k7_lopclset(A))) ) )).

fof(t26_lopclset,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & v10_lattices(B)
        & v17_lattices(B)
        & ~ v3_realset2(B)
        & l3_lattices(B) )
     => ( r2_hidden(A,k10_lopclset(B))
      <=> ? [C] :
            ( m1_subset_1(C,u1_struct_0(B))
            & k8_funct_2(u1_struct_0(B),k1_zfmisc_1(k7_lopclset(B)),k9_lopclset(B),C) = A ) ) ) )).

fof(t27_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_filter_0(C,A)
             => ~ ( v1_filter_0(C,A)
                  & ~ r2_hidden(C,k8_funct_2(u1_struct_0(A),k1_zfmisc_1(k7_lopclset(A)),k9_lopclset(A),B))
                  & r2_hidden(B,C) ) ) ) ) )).

fof(t28_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k4_xboole_0(k7_lopclset(A),k8_funct_2(u1_struct_0(A),k1_zfmisc_1(k7_lopclset(A)),k9_lopclset(A),B)) = k8_funct_2(u1_struct_0(A),k1_zfmisc_1(k7_lopclset(A)),k9_lopclset(A),k7_lattices(A,B)) ) ) )).

fof(d9_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => k12_lopclset(A) = k6_lopclset(k11_lopclset(A)) ) )).

fof(t29_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => v2_funct_1(k9_lopclset(A)) ) )).

fof(t30_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => k3_tarski(k10_lopclset(A)) = k7_lopclset(A) ) )).

fof(t31_lopclset,axiom,(
    $true )).

fof(t32_lopclset,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ~ ! [B] :
            ( m1_subset_1(B,k5_finsub_1(A))
           => v1_xboole_0(B) ) ) )).

fof(t33_lopclset,axiom,(
    $true )).

fof(t34_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ~ ( r2_hidden(k5_lattices(A),k3_filter_0(A,B))
              & ! [C] :
                  ( ( ~ v1_xboole_0(C)
                    & m1_subset_1(C,k5_finsub_1(u1_struct_0(A))) )
                 => ~ ( r1_tarski(C,B)
                      & k2_lattice4(A,C) = k5_lattices(A) ) ) ) ) ) )).

fof(t35_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v13_lattices(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m1_filter_0(B,A)
         => ~ ( v1_filter_0(B,A)
              & r2_hidden(k5_lattices(A),B) ) ) ) )).

fof(t36_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => k8_funct_2(u1_struct_0(A),k1_zfmisc_1(k7_lopclset(A)),k9_lopclset(A),k5_lattices(A)) = k1_xboole_0 ) )).

fof(t37_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => k8_funct_2(u1_struct_0(A),k1_zfmisc_1(k7_lopclset(A)),k9_lopclset(A),k6_lattices(A)) = k7_lopclset(A) ) )).

fof(t38_lopclset,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & v10_lattices(B)
        & v17_lattices(B)
        & ~ v3_realset2(B)
        & l3_lattices(B) )
     => ~ ( k7_lopclset(B) = k3_tarski(A)
          & m1_subset_1(A,k1_zfmisc_1(k10_lopclset(B)))
          & ! [C] :
              ( m1_subset_1(C,k5_finsub_1(A))
             => k7_lopclset(B) != k3_tarski(C) ) ) ) )).

fof(t39_lopclset,axiom,(
    $true )).

fof(t40_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => k10_lopclset(A) = k1_lopclset(k11_lopclset(A)) ) )).

fof(t41_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => k2_relat_1(k13_lopclset(A)) = u1_struct_0(k12_lopclset(A)) ) )).

fof(t42_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => v3_lattice4(k13_lopclset(A),A,k12_lopclset(A)) ) )).

fof(t43_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => r1_filter_1(A,k12_lopclset(A)) ) )).

fof(t44_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => ? [B] :
          ( ~ v3_struct_0(B)
          & v2_pre_topc(B)
          & l1_pre_topc(B)
          & r1_filter_1(A,k6_lopclset(B)) ) ) )).

fof(dt_k1_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A) )
     => m1_subset_1(k1_lopclset(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) )).

fof(dt_k2_lopclset,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & m1_subset_1(B,k1_lopclset(A))
        & m1_subset_1(C,k1_lopclset(A)) )
     => m2_subset_1(k2_lopclset(A,B,C),k1_zfmisc_1(u1_struct_0(A)),k1_lopclset(A)) ) )).

fof(commutativity_k2_lopclset,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & m1_subset_1(B,k1_lopclset(A))
        & m1_subset_1(C,k1_lopclset(A)) )
     => k2_lopclset(A,B,C) = k2_lopclset(A,C,B) ) )).

fof(idempotence_k2_lopclset,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & m1_subset_1(B,k1_lopclset(A))
        & m1_subset_1(C,k1_lopclset(A)) )
     => k2_lopclset(A,B,B) = B ) )).

fof(redefinition_k2_lopclset,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & m1_subset_1(B,k1_lopclset(A))
        & m1_subset_1(C,k1_lopclset(A)) )
     => k2_lopclset(A,B,C) = k2_xboole_0(B,C) ) )).

fof(dt_k3_lopclset,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & m1_subset_1(B,k1_lopclset(A))
        & m1_subset_1(C,k1_lopclset(A)) )
     => m2_subset_1(k3_lopclset(A,B,C),k1_zfmisc_1(u1_struct_0(A)),k1_lopclset(A)) ) )).

fof(commutativity_k3_lopclset,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & m1_subset_1(B,k1_lopclset(A))
        & m1_subset_1(C,k1_lopclset(A)) )
     => k3_lopclset(A,B,C) = k3_lopclset(A,C,B) ) )).

fof(idempotence_k3_lopclset,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & m1_subset_1(B,k1_lopclset(A))
        & m1_subset_1(C,k1_lopclset(A)) )
     => k3_lopclset(A,B,B) = B ) )).

fof(redefinition_k3_lopclset,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & m1_subset_1(B,k1_lopclset(A))
        & m1_subset_1(C,k1_lopclset(A)) )
     => k3_lopclset(A,B,C) = k3_xboole_0(B,C) ) )).

fof(dt_k4_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v1_funct_1(k4_lopclset(A))
        & v1_funct_2(k4_lopclset(A),k2_zfmisc_1(k1_lopclset(A),k1_lopclset(A)),k1_lopclset(A))
        & m2_relset_1(k4_lopclset(A),k2_zfmisc_1(k1_lopclset(A),k1_lopclset(A)),k1_lopclset(A)) ) ) )).

fof(dt_k5_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v1_funct_1(k5_lopclset(A))
        & v1_funct_2(k5_lopclset(A),k2_zfmisc_1(k1_lopclset(A),k1_lopclset(A)),k1_lopclset(A))
        & m2_relset_1(k5_lopclset(A),k2_zfmisc_1(k1_lopclset(A),k1_lopclset(A)),k1_lopclset(A)) ) ) )).

fof(dt_k6_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( ~ v3_struct_0(k6_lopclset(A))
        & v10_lattices(k6_lopclset(A))
        & l3_lattices(k6_lopclset(A)) ) ) )).

fof(dt_k7_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => m1_subset_1(k7_lopclset(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) )).

fof(dt_k8_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => ( v1_relat_1(k8_lopclset(A))
        & v1_funct_1(k8_lopclset(A)) ) ) )).

fof(dt_k9_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => ( v1_funct_1(k9_lopclset(A))
        & v1_funct_2(k9_lopclset(A),u1_struct_0(A),k1_zfmisc_1(k7_lopclset(A)))
        & m2_relset_1(k9_lopclset(A),u1_struct_0(A),k1_zfmisc_1(k7_lopclset(A))) ) ) )).

fof(redefinition_k9_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => k9_lopclset(A) = k8_lopclset(A) ) )).

fof(dt_k10_lopclset,axiom,(
    $true )).

fof(dt_k11_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => ( v1_pre_topc(k11_lopclset(A))
        & v2_pre_topc(k11_lopclset(A))
        & l1_pre_topc(k11_lopclset(A)) ) ) )).

fof(dt_k12_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => ( ~ v3_struct_0(k12_lopclset(A))
        & v10_lattices(k12_lopclset(A))
        & l3_lattices(k12_lopclset(A)) ) ) )).

fof(dt_k13_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => m1_lattice4(k13_lopclset(A),A,k12_lopclset(A)) ) )).

fof(redefinition_k13_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => k13_lopclset(A) = k8_lopclset(A) ) )).

fof(d1_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A) )
     => k1_lopclset(A) = a_1_0_lopclset(A) ) )).

fof(d5_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => k7_lopclset(A) = a_1_1_lopclset(A) ) )).

fof(t19_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => r1_tarski(a_2_0_lopclset(A,B),k7_lopclset(A)) ) ) )).

fof(d6_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( B = k8_lopclset(A)
          <=> ( k1_relat_1(B) = u1_struct_0(A)
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => k1_funct_1(B,C) = a_2_0_lopclset(A,C) ) ) ) ) ) )).

fof(d8_lopclset,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v17_lattices(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => ! [B] :
          ( ( v1_pre_topc(B)
            & v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ( B = k11_lopclset(A)
          <=> ( u1_struct_0(B) = k7_lopclset(A)
              & u1_pre_topc(B) = a_1_2_lopclset(A) ) ) ) ) )).

fof(fraenkel_a_1_0_lopclset,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l1_pre_topc(B) )
     => ( r2_hidden(A,a_1_0_lopclset(B))
      <=> ? [C] :
            ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
            & A = C
            & v3_pre_topc(C,B)
            & v4_pre_topc(C,B) ) ) ) )).

fof(fraenkel_a_1_1_lopclset,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & v10_lattices(B)
        & v17_lattices(B)
        & ~ v3_realset2(B)
        & l3_lattices(B) )
     => ( r2_hidden(A,a_1_1_lopclset(B))
      <=> ? [C] :
            ( m1_filter_0(C,B)
            & A = C
            & v1_filter_0(C,B) ) ) ) )).

fof(fraenkel_a_2_0_lopclset,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & v10_lattices(B)
        & v17_lattices(B)
        & ~ v3_realset2(B)
        & l3_lattices(B)
        & m1_subset_1(C,u1_struct_0(B)) )
     => ( r2_hidden(A,a_2_0_lopclset(B,C))
      <=> ? [D] :
            ( m1_filter_0(D,B)
            & A = D
            & v1_filter_0(D,B)
            & r2_hidden(C,D) ) ) ) )).

fof(fraenkel_a_1_2_lopclset,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & v10_lattices(B)
        & v17_lattices(B)
        & ~ v3_realset2(B)
        & l3_lattices(B) )
     => ( r2_hidden(A,a_1_2_lopclset(B))
      <=> ? [C] :
            ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(k7_lopclset(B))))
            & A = k5_setfam_1(k7_lopclset(B),C)
            & r1_tarski(C,k10_lopclset(B)) ) ) ) )).
%------------------------------------------------------------------------------