SET007 Axioms: SET007+377.ax
%------------------------------------------------------------------------------
% File : SET007+377 : TPTP v8.2.0. 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 unt; 0 def)
% Number of atoms : 548 ( 50 equ)
% Maximal formula atoms : 17 ( 6 avg)
% Number of connectives : 603 ( 139 ~; 1 |; 323 &)
% ( 13 <=>; 127 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 8 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 40 ( 38 usr; 1 prp; 0-3 aty)
% Number of functors : 46 ( 46 usr; 1 con; 0-6 aty)
% Number of variables : 156 ( 146 !; 10 ?)
% 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)) ) ) ) ).
%------------------------------------------------------------------------------