TPTP Problem File: LAT290+1.p
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%------------------------------------------------------------------------------
% File : LAT290+1 : TPTP v8.2.0. Released v3.4.0.
% Domain : Lattice Theory
% Problem : Representation Theorem for Boolean Algebras T30
% Version : [Urb08] axioms : Especial.
% English :
% Refs : [Wal93] Walijewski (1993), Representation Theorem for Boolean
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : t30_lopclset [Urb08]
% Status : Theorem
% Rating : 0.97 v7.1.0, 0.96 v7.0.0, 1.00 v6.4.0, 0.96 v6.2.0, 1.00 v3.4.0
% Syntax : Number of formulae : 146 ( 25 unt; 0 def)
% Number of atoms : 570 ( 34 equ)
% Maximal formula atoms : 17 ( 3 avg)
% Number of connectives : 519 ( 95 ~; 1 |; 272 &)
% ( 11 <=>; 140 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 5 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 54 ( 52 usr; 1 prp; 0-3 aty)
% Number of functors : 23 ( 23 usr; 1 con; 0-3 aty)
% Number of variables : 242 ( 216 !; 26 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Normal version: includes the axioms (which may be theorems from
% other articles) and background that are possibly necessary.
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : The problem encoding is based on set theory.
%------------------------------------------------------------------------------
fof(t30_lopclset,conjecture,
! [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(abstractness_v1_pre_topc,axiom,
! [A] :
( l1_pre_topc(A)
=> ( v1_pre_topc(A)
=> A = g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) ) ) ).
fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( r2_hidden(A,B)
=> ~ r2_hidden(B,A) ) ).
fof(cc10_membered,axiom,
! [A] :
( v1_membered(A)
=> ! [B] :
( m1_subset_1(B,A)
=> v1_xcmplx_0(B) ) ) ).
fof(cc11_membered,axiom,
! [A] :
( v2_membered(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ( v1_xcmplx_0(B)
& v1_xreal_0(B) ) ) ) ).
fof(cc12_membered,axiom,
! [A] :
( v3_membered(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ( v1_xcmplx_0(B)
& v1_xreal_0(B)
& v1_rat_1(B) ) ) ) ).
fof(cc13_membered,axiom,
! [A] :
( v4_membered(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ( v1_xcmplx_0(B)
& v1_xreal_0(B)
& v1_int_1(B)
& v1_rat_1(B) ) ) ) ).
fof(cc14_membered,axiom,
! [A] :
( v5_membered(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ( v1_xcmplx_0(B)
& v4_ordinal2(B)
& v1_xreal_0(B)
& v1_int_1(B)
& v1_rat_1(B) ) ) ) ).
fof(cc15_membered,axiom,
! [A] :
( v1_xboole_0(A)
=> ( v1_membered(A)
& v2_membered(A)
& v3_membered(A)
& v4_membered(A)
& v5_membered(A) ) ) ).
fof(cc16_membered,axiom,
! [A] :
( v1_membered(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> v1_membered(B) ) ) ).
fof(cc17_membered,axiom,
! [A] :
( v2_membered(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( v1_membered(B)
& v2_membered(B) ) ) ) ).
fof(cc18_membered,axiom,
! [A] :
( v3_membered(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( v1_membered(B)
& v2_membered(B)
& v3_membered(B) ) ) ) ).
fof(cc19_membered,axiom,
! [A] :
( v4_membered(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( v1_membered(B)
& v2_membered(B)
& v3_membered(B)
& v4_membered(B) ) ) ) ).
fof(cc1_finset_1,axiom,
! [A] :
( v1_xboole_0(A)
=> v1_finset_1(A) ) ).
fof(cc1_finsub_1,axiom,
! [A] :
( v4_finsub_1(A)
=> ( v1_finsub_1(A)
& v3_finsub_1(A) ) ) ).
fof(cc1_funct_2,axiom,
! [A,B,C] :
( m1_relset_1(C,A,B)
=> ( ( v1_funct_1(C)
& v1_partfun1(C,A,B) )
=> ( v1_funct_1(C)
& v1_funct_2(C,A,B) ) ) ) ).
fof(cc1_lattices,axiom,
! [A] :
( l3_lattices(A)
=> ( ( ~ v3_struct_0(A)
& v10_lattices(A) )
=> ( ~ v3_struct_0(A)
& v4_lattices(A)
& v5_lattices(A)
& v6_lattices(A)
& v7_lattices(A)
& v8_lattices(A)
& v9_lattices(A) ) ) ) ).
fof(cc1_membered,axiom,
! [A] :
( v5_membered(A)
=> v4_membered(A) ) ).
fof(cc1_relset_1,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> v1_relat_1(C) ) ).
fof(cc20_membered,axiom,
! [A] :
( v5_membered(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( v1_membered(B)
& v2_membered(B)
& v3_membered(B)
& v4_membered(B)
& v5_membered(B) ) ) ) ).
fof(cc2_finset_1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> v1_finset_1(B) ) ) ).
fof(cc2_finsub_1,axiom,
! [A] :
( ( v1_finsub_1(A)
& v3_finsub_1(A) )
=> v4_finsub_1(A) ) ).
fof(cc2_lattices,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) )
=> ( ~ v3_struct_0(A)
& v10_lattices(A) ) ) ) ).
fof(cc2_membered,axiom,
! [A] :
( v4_membered(A)
=> v3_membered(A) ) ).
fof(cc3_lattices,axiom,
! [A] :
( l3_lattices(A)
=> ( ( ~ v3_struct_0(A)
& v13_lattices(A)
& v14_lattices(A) )
=> ( ~ v3_struct_0(A)
& v15_lattices(A) ) ) ) ).
fof(cc3_membered,axiom,
! [A] :
( v3_membered(A)
=> v2_membered(A) ) ).
fof(cc4_lattices,axiom,
! [A] :
( l3_lattices(A)
=> ( ( ~ v3_struct_0(A)
& v15_lattices(A) )
=> ( ~ v3_struct_0(A)
& v13_lattices(A)
& v14_lattices(A) ) ) ) ).
fof(cc4_membered,axiom,
! [A] :
( v2_membered(A)
=> v1_membered(A) ) ).
fof(cc5_funct_2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_relset_1(C,A,B)
=> ( ( v1_funct_1(C)
& v1_funct_2(C,A,B) )
=> ( v1_funct_1(C)
& v1_partfun1(C,A,B)
& v1_funct_2(C,A,B) ) ) ) ) ).
fof(cc5_lattices,axiom,
! [A] :
( l3_lattices(A)
=> ( ( ~ v3_struct_0(A)
& v17_lattices(A) )
=> ( ~ v3_struct_0(A)
& v11_lattices(A)
& v13_lattices(A)
& v14_lattices(A)
& v15_lattices(A)
& v16_lattices(A) ) ) ) ).
fof(cc6_funct_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B) )
=> ! [C] :
( m1_relset_1(C,A,B)
=> ( ( v1_funct_1(C)
& v1_funct_2(C,A,B) )
=> ( v1_funct_1(C)
& ~ v1_xboole_0(C)
& v1_partfun1(C,A,B)
& v1_funct_2(C,A,B) ) ) ) ) ).
fof(cc6_lattices,axiom,
! [A] :
( l3_lattices(A)
=> ( ( ~ v3_struct_0(A)
& v11_lattices(A)
& v15_lattices(A)
& v16_lattices(A) )
=> ( ~ v3_struct_0(A)
& v17_lattices(A) ) ) ) ).
fof(cc7_lattices,axiom,
! [A] :
( l3_lattices(A)
=> ( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_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)
& v12_lattices(A) ) ) ) ).
fof(commutativity_k1_finsub_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> k1_finsub_1(A,B,C) = k1_finsub_1(A,C,B) ) ).
fof(commutativity_k2_xboole_0,axiom,
! [A,B] : k2_xboole_0(A,B) = k2_xboole_0(B,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(d10_xboole_0,axiom,
! [A,B] :
( A = B
<=> ( r1_tarski(A,B)
& r1_tarski(B,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(d5_pre_topc,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v3_pre_topc(B,A)
<=> r2_hidden(B,u1_pre_topc(A)) ) ) ) ).
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(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(dt_g1_pre_topc,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( v1_pre_topc(g1_pre_topc(A,B))
& l1_pre_topc(g1_pre_topc(A,B)) ) ) ).
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_k1_finsub_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> m1_subset_1(k1_finsub_1(A,B,C),A) ) ).
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_k1_xboole_0,axiom,
$true ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_pre_topc,axiom,
! [A] :
( l1_struct_0(A)
=> m1_subset_1(k2_pre_topc(A),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k2_relat_1,axiom,
$true ).
fof(dt_k2_xboole_0,axiom,
$true ).
fof(dt_k2_zfmisc_1,axiom,
$true ).
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(dt_k3_tarski,axiom,
$true ).
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(dt_k5_setfam_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> m1_subset_1(k5_setfam_1(A,B),k1_zfmisc_1(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(dt_l1_lattices,axiom,
! [A] :
( l1_lattices(A)
=> l1_struct_0(A) ) ).
fof(dt_l1_pre_topc,axiom,
! [A] :
( l1_pre_topc(A)
=> l1_struct_0(A) ) ).
fof(dt_l1_struct_0,axiom,
$true ).
fof(dt_l2_lattices,axiom,
! [A] :
( l2_lattices(A)
=> l1_struct_0(A) ) ).
fof(dt_l3_lattices,axiom,
! [A] :
( l3_lattices(A)
=> ( l1_lattices(A)
& l2_lattices(A) ) ) ).
fof(dt_m1_filter_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_filter_0(B,A)
=> ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) ) ) ).
fof(dt_m1_relset_1,axiom,
$true ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(dt_m2_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
=> m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ).
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(dt_u1_pre_topc,axiom,
! [A] :
( l1_pre_topc(A)
=> m1_subset_1(u1_pre_topc(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ).
fof(dt_u1_struct_0,axiom,
$true ).
fof(existence_l1_lattices,axiom,
? [A] : l1_lattices(A) ).
fof(existence_l1_pre_topc,axiom,
? [A] : l1_pre_topc(A) ).
fof(existence_l1_struct_0,axiom,
? [A] : l1_struct_0(A) ).
fof(existence_l2_lattices,axiom,
? [A] : l2_lattices(A) ).
fof(existence_l3_lattices,axiom,
? [A] : l3_lattices(A) ).
fof(existence_m1_filter_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ? [B] : m1_filter_0(B,A) ) ).
fof(existence_m1_relset_1,axiom,
! [A,B] :
? [C] : m1_relset_1(C,A,B) ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : m1_subset_1(B,A) ).
fof(existence_m2_relset_1,axiom,
! [A,B] :
? [C] : m2_relset_1(C,A,B) ).
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(fc14_finset_1,axiom,
! [A,B] :
( ( v1_finset_1(A)
& v1_finset_1(B) )
=> v1_finset_1(k2_zfmisc_1(A,B)) ) ).
fof(fc1_finsub_1,axiom,
! [A] :
( ~ v1_xboole_0(k1_zfmisc_1(A))
& v1_finsub_1(k1_zfmisc_1(A))
& v3_finsub_1(k1_zfmisc_1(A))
& v4_finsub_1(k1_zfmisc_1(A)) ) ).
fof(fc1_lopclset,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ~ v1_xboole_0(k1_lopclset(A)) ) ).
fof(fc1_struct_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ~ v1_xboole_0(u1_struct_0(A)) ) ).
fof(fc22_membered,axiom,
! [A,B] :
( ( v1_membered(A)
& v1_membered(B) )
=> v1_membered(k2_xboole_0(A,B)) ) ).
fof(fc23_membered,axiom,
! [A,B] :
( ( v2_membered(A)
& v2_membered(B) )
=> ( v1_membered(k2_xboole_0(A,B))
& v2_membered(k2_xboole_0(A,B)) ) ) ).
fof(fc24_membered,axiom,
! [A,B] :
( ( v3_membered(A)
& v3_membered(B) )
=> ( v1_membered(k2_xboole_0(A,B))
& v2_membered(k2_xboole_0(A,B))
& v3_membered(k2_xboole_0(A,B)) ) ) ).
fof(fc25_membered,axiom,
! [A,B] :
( ( v4_membered(A)
& v4_membered(B) )
=> ( v1_membered(k2_xboole_0(A,B))
& v2_membered(k2_xboole_0(A,B))
& v3_membered(k2_xboole_0(A,B))
& v4_membered(k2_xboole_0(A,B)) ) ) ).
fof(fc26_membered,axiom,
! [A,B] :
( ( v5_membered(A)
& v5_membered(B) )
=> ( v1_membered(k2_xboole_0(A,B))
& v2_membered(k2_xboole_0(A,B))
& v3_membered(k2_xboole_0(A,B))
& v4_membered(k2_xboole_0(A,B))
& v5_membered(k2_xboole_0(A,B)) ) ) ).
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(fc2_pre_topc,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ~ v1_xboole_0(k2_pre_topc(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(fc5_pre_topc,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> v4_pre_topc(k2_pre_topc(A),A) ) ).
fof(fc6_membered,axiom,
( v1_xboole_0(k1_xboole_0)
& v1_membered(k1_xboole_0)
& v2_membered(k1_xboole_0)
& v3_membered(k1_xboole_0)
& v4_membered(k1_xboole_0)
& v5_membered(k1_xboole_0) ) ).
fof(fc9_finset_1,axiom,
! [A,B] :
( ( v1_finset_1(A)
& v1_finset_1(B) )
=> v1_finset_1(k2_xboole_0(A,B)) ) ).
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_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)) ) ) ) ).
fof(free_g1_pre_topc,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ! [C,D] :
( g1_pre_topc(A,B) = g1_pre_topc(C,D)
=> ( A = C
& B = D ) ) ) ).
fof(idempotence_k1_finsub_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> k1_finsub_1(A,B,B) = B ) ).
fof(idempotence_k2_xboole_0,axiom,
! [A,B] : k2_xboole_0(A,A) = A ).
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(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(rc1_finset_1,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_finset_1(A) ) ).
fof(rc1_funct_2,axiom,
! [A,B] :
? [C] :
( m1_relset_1(C,A,B)
& v1_relat_1(C)
& v1_funct_1(C)
& v1_funct_2(C,A,B) ) ).
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(rc1_membered,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_membered(A)
& v2_membered(A)
& v3_membered(A)
& v4_membered(A)
& v5_membered(A) ) ).
fof(rc1_pre_topc,axiom,
? [A] :
( l1_pre_topc(A)
& v1_pre_topc(A) ) ).
fof(rc2_partfun1,axiom,
! [A,B] :
? [C] :
( m1_relset_1(C,A,B)
& v1_relat_1(C)
& v1_funct_1(C) ) ).
fof(rc2_pre_topc,axiom,
? [A] :
( l1_pre_topc(A)
& ~ v3_struct_0(A)
& v1_pre_topc(A)
& v2_pre_topc(A) ) ).
fof(rc3_finset_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& ~ v1_xboole_0(B)
& v1_finset_1(B) ) ) ).
fof(rc3_struct_0,axiom,
? [A] :
( l1_struct_0(A)
& ~ v3_struct_0(A) ) ).
fof(rc4_finset_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& ~ v1_xboole_0(B)
& v1_finset_1(B) ) ) ).
fof(rc5_struct_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(B) ) ) ).
fof(rc6_pre_topc,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& v4_pre_topc(B,A) ) ) ).
fof(rc7_pre_topc,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)))
& ~ v1_xboole_0(B)
& v4_pre_topc(B,A) ) ) ).
fof(redefinition_k1_finsub_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> k1_finsub_1(A,B,C) = k2_xboole_0(B,C) ) ).
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(redefinition_k5_setfam_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> k5_setfam_1(A,B) = k3_tarski(B) ) ).
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(redefinition_m2_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
<=> m1_relset_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(reflexivity_r1_tarski,axiom,
! [A,B] : r1_tarski(A,A) ).
fof(t12_pre_topc,axiom,
! [A] :
( l1_struct_0(A)
=> k2_pre_topc(A) = u1_struct_0(A) ) ).
fof(t18_pre_topc,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k4_subset_1(u1_struct_0(A),B,k3_subset_1(u1_struct_0(A),B)) = k2_pre_topc(A) ) ) ).
fof(t1_boole,axiom,
! [A] : k2_xboole_0(A,k1_xboole_0) = A ).
fof(t1_subset,axiom,
! [A,B] :
( r2_hidden(A,B)
=> m1_subset_1(A,B) ) ).
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(t29_tops_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v4_pre_topc(B,A)
<=> v3_pre_topc(k3_subset_1(u1_struct_0(A),B),A) ) ) ) ).
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(t2_subset,axiom,
! [A,B] :
( m1_subset_1(A,B)
=> ( v1_xboole_0(B)
| r2_hidden(A,B) ) ) ).
fof(t2_tarski,axiom,
! [A,B] :
( ! [C] :
( r2_hidden(C,A)
<=> r2_hidden(C,B) )
=> A = B ) ).
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(t3_subset,axiom,
! [A,B] :
( m1_subset_1(A,k1_zfmisc_1(B))
<=> r1_tarski(A,B) ) ).
fof(t4_subset,axiom,
! [A,B,C] :
( ( r2_hidden(A,B)
& m1_subset_1(B,k1_zfmisc_1(C)) )
=> m1_subset_1(A,C) ) ).
fof(t5_subset,axiom,
! [A,B,C] :
~ ( r2_hidden(A,B)
& m1_subset_1(B,k1_zfmisc_1(C))
& v1_xboole_0(C) ) ).
fof(t6_boole,axiom,
! [A] :
( v1_xboole_0(A)
=> A = k1_xboole_0 ) ).
fof(t7_boole,axiom,
! [A,B] :
~ ( r2_hidden(A,B)
& v1_xboole_0(B) ) ).
fof(t8_boole,axiom,
! [A,B] :
~ ( v1_xboole_0(A)
& A != B
& v1_xboole_0(B) ) ).
fof(t8_xboole_1,axiom,
! [A,B,C] :
( ( r1_tarski(A,B)
& r1_tarski(C,B) )
=> r1_tarski(k2_xboole_0(A,C),B) ) ).
fof(t95_zfmisc_1,axiom,
! [A,B] :
( r1_tarski(A,B)
=> r1_tarski(k3_tarski(A),k3_tarski(B)) ) ).
fof(t96_zfmisc_1,axiom,
! [A,B] : k3_tarski(k2_xboole_0(A,B)) = k2_xboole_0(k3_tarski(A),k3_tarski(B)) ).
fof(t99_zfmisc_1,axiom,
! [A] : k3_tarski(k1_zfmisc_1(A)) = A ).
%------------------------------------------------------------------------------