SET007 Axioms: SET007+376.ax
%------------------------------------------------------------------------------
% File : SET007+376 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Representation Theorem for Heyting Lattices
% 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 : openlatt [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 99 ( 2 unt; 0 def)
% Number of atoms : 698 ( 63 equ)
% Maximal formula atoms : 18 ( 7 avg)
% Number of connectives : 737 ( 138 ~; 0 |; 418 &)
% ( 19 <=>; 162 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 8 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 44 ( 42 usr; 1 prp; 0-3 aty)
% Number of functors : 47 ( 47 usr; 1 con; 0-6 aty)
% Number of variables : 209 ( 202 !; 7 ?)
% 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(cc1_openlatt,axiom,
! [A] :
( l3_lattices(A)
=> ( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v13_lattices(A)
& v1_lattice2(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)
& v1_lattice2(A)
& v3_filter_0(A) ) ) ) ).
fof(cc2_openlatt,axiom,
! [A] :
( l3_lattices(A)
=> ( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v3_filter_0(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)
& v14_lattices(A) ) ) ) ).
fof(fc1_openlatt,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ~ v1_xboole_0(k1_openlatt(A)) ) ).
fof(fc2_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ~ v3_struct_0(k6_openlatt(A))
& v4_lattices(k6_openlatt(A))
& v5_lattices(k6_openlatt(A))
& v6_lattices(k6_openlatt(A))
& v7_lattices(k6_openlatt(A))
& v8_lattices(k6_openlatt(A))
& v9_lattices(k6_openlatt(A))
& v10_lattices(k6_openlatt(A))
& v11_lattices(k6_openlatt(A))
& v12_lattices(k6_openlatt(A))
& v13_lattices(k6_openlatt(A))
& v14_lattices(k6_openlatt(A))
& v15_lattices(k6_openlatt(A))
& v1_lattice2(k6_openlatt(A))
& v3_filter_0(k6_openlatt(A)) ) ) ).
fof(fc3_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> ~ v1_xboole_0(k9_openlatt(A)) ) ).
fof(fc4_openlatt,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ~ v1_xboole_0(k10_openlatt(A,B)) ) ).
fof(rc1_openlatt,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)
& v1_lattice2(A)
& v3_filter_0(A)
& ~ v3_realset2(A) ) ).
fof(fc5_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v1_lattice2(A)
& ~ v3_realset2(A)
& l3_lattices(A) )
=> ~ v1_xboole_0(k7_openlatt(A)) ) ).
fof(fc6_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v1_lattice2(A)
& ~ v3_realset2(A)
& l3_lattices(A) )
=> ( ~ v3_struct_0(k17_openlatt(A))
& v1_pre_topc(k17_openlatt(A))
& v2_pre_topc(k17_openlatt(A)) ) ) ).
fof(t1_openlatt,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> r1_tarski(k5_subset_1(u1_struct_0(A),B,k1_tops_1(A,k4_subset_1(u1_struct_0(A),k3_subset_1(u1_struct_0(A),B),C))),C) ) ) ) ).
fof(t2_openlatt,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v3_pre_topc(D,A)
& r1_tarski(k5_subset_1(u1_struct_0(A),B,D),C) )
=> r1_tarski(D,k1_tops_1(A,k4_subset_1(u1_struct_0(A),k3_subset_1(u1_struct_0(A),B),C))) ) ) ) ) ) ).
fof(d1_openlatt,axiom,
! [A] :
( l1_pre_topc(A)
=> k1_openlatt(A) = u1_pre_topc(A) ) ).
fof(t3_openlatt,axiom,
! [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)
<=> r2_hidden(B,k1_openlatt(A)) ) ) ) ).
fof(d2_openlatt,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_openlatt(A),k1_openlatt(A)),k1_openlatt(A))
& m2_relset_1(B,k2_zfmisc_1(k1_openlatt(A),k1_openlatt(A)),k1_openlatt(A)) )
=> ( B = k4_openlatt(A)
<=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(u1_struct_0(A)),k1_openlatt(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(u1_struct_0(A)),k1_openlatt(A))
=> k2_binop_1(k1_openlatt(A),k1_openlatt(A),k1_openlatt(A),B,C,D) = k2_openlatt(A,C,D) ) ) ) ) ) ).
fof(d3_openlatt,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_openlatt(A),k1_openlatt(A)),k1_openlatt(A))
& m2_relset_1(B,k2_zfmisc_1(k1_openlatt(A),k1_openlatt(A)),k1_openlatt(A)) )
=> ( B = k5_openlatt(A)
<=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(u1_struct_0(A)),k1_openlatt(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(u1_struct_0(A)),k1_openlatt(A))
=> k2_binop_1(k1_openlatt(A),k1_openlatt(A),k1_openlatt(A),B,C,D) = k3_openlatt(A,C,D) ) ) ) ) ) ).
fof(t4_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ~ v3_struct_0(g3_lattices(k1_openlatt(A),k4_openlatt(A),k5_openlatt(A)))
& v10_lattices(g3_lattices(k1_openlatt(A),k4_openlatt(A),k5_openlatt(A)))
& l3_lattices(g3_lattices(k1_openlatt(A),k4_openlatt(A),k5_openlatt(A))) ) ) ).
fof(d4_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> k6_openlatt(A) = g3_lattices(k1_openlatt(A),k4_openlatt(A),k5_openlatt(A)) ) ).
fof(t5_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> u1_struct_0(k6_openlatt(A)) = k1_openlatt(A) ) ).
fof(t6_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k6_openlatt(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k6_openlatt(A)))
=> ( k3_lattices(k6_openlatt(A),B,C) = k2_xboole_0(B,C)
& k4_lattices(k6_openlatt(A),B,C) = k3_xboole_0(B,C) ) ) ) ) ).
fof(t7_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k6_openlatt(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k6_openlatt(A)))
=> ( r3_lattices(k6_openlatt(A),B,C)
<=> r1_tarski(B,C) ) ) ) ) ).
fof(t8_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k6_openlatt(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k6_openlatt(A)))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(u1_struct_0(A)),k1_openlatt(A))
=> ! [E] :
( m2_subset_1(E,k1_zfmisc_1(u1_struct_0(A)),k1_openlatt(A))
=> ( ( B = D
& C = E )
=> ( r3_lattices(k6_openlatt(A),B,C)
<=> r1_tarski(D,E) ) ) ) ) ) ) ) ).
fof(t9_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> v3_filter_0(k6_openlatt(A)) ) ).
fof(t10_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v13_lattices(k6_openlatt(A))
& k5_lattices(k6_openlatt(A)) = k1_xboole_0 ) ) ).
fof(t11_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> k6_lattices(k6_openlatt(A)) = u1_struct_0(A) ) ).
fof(t12_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_filter_0(B,A)
=> ( r2_hidden(B,k7_openlatt(A))
<=> ( B != u1_struct_0(A)
& v2_filter_0(B,A) ) ) ) ) ).
fof(t13_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_filter_0(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(B,k1_funct_1(k8_openlatt(A),C))
<=> ( r2_hidden(B,k7_openlatt(A))
& r2_hidden(C,B) ) ) ) ) ) ).
fof(t14_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( r2_hidden(C,k1_funct_1(k8_openlatt(A),B))
<=> ? [D] :
( m1_filter_0(D,A)
& D = C
& D != u1_struct_0(A)
& v2_filter_0(D,A)
& r2_hidden(B,D) ) ) ) ) ).
fof(d7_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> k9_openlatt(A) = k2_relat_1(k8_openlatt(A)) ) ).
fof(t15_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( r2_hidden(B,k9_openlatt(A))
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& B = k1_funct_1(k8_openlatt(A),C) ) ) ) ).
fof(t16_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(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_openlatt(A),k3_lattices(A,B,C)) = k2_xboole_0(k1_funct_1(k8_openlatt(A),B),k1_funct_1(k8_openlatt(A),C)) ) ) ) ).
fof(t17_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(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_openlatt(A),k4_lattices(A,B,C)) = k3_xboole_0(k1_funct_1(k8_openlatt(A),B),k1_funct_1(k8_openlatt(A),C)) ) ) ) ).
fof(t18_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( r2_hidden(C,k10_openlatt(A,B))
<=> ( m1_filter_0(C,A)
& r2_hidden(B,C) ) ) ) ) ).
fof(t19_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( r2_hidden(D,k6_subset_1(k1_zfmisc_1(u1_struct_0(A)),k10_openlatt(A,B),k10_openlatt(A,C)))
=> ( m1_filter_0(D,A)
& r2_hidden(B,D)
& ~ r2_hidden(C,D) ) ) ) ) ) ).
fof(t20_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
~ ( D != k1_xboole_0
& r1_tarski(D,k6_subset_1(k1_zfmisc_1(u1_struct_0(A)),k10_openlatt(A,B),k10_openlatt(A,C)))
& v6_ordinal1(D)
& ! [E] :
~ ( r2_hidden(E,k6_subset_1(k1_zfmisc_1(u1_struct_0(A)),k10_openlatt(A,B),k10_openlatt(A,C)))
& ! [F] :
( r2_hidden(F,D)
=> r1_tarski(F,E) ) ) ) ) ) ) ).
fof(t21_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ~ r3_lattices(A,B,C)
=> r2_hidden(k2_filter_0(A,B),k6_subset_1(k1_zfmisc_1(u1_struct_0(A)),k10_openlatt(A,B),k10_openlatt(A,C))) ) ) ) ) ).
fof(t22_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( ~ r3_lattices(A,B,C)
& ! [D] :
( m1_filter_0(D,A)
=> ~ ( r2_hidden(D,k7_openlatt(A))
& ~ r2_hidden(C,D)
& r2_hidden(B,D) ) ) ) ) ) ) ).
fof(t23_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( B != C
& ! [D] :
( m1_filter_0(D,A)
=> ~ r2_hidden(D,k7_openlatt(A)) ) ) ) ) ) ).
fof(t24_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( B != C
& k1_funct_1(k8_openlatt(A),B) = k1_funct_1(k8_openlatt(A),C) ) ) ) ) ).
fof(t25_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> v2_funct_1(k8_openlatt(A)) ) ).
fof(d9_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(k9_openlatt(A),k9_openlatt(A)),k9_openlatt(A))
& m2_relset_1(B,k2_zfmisc_1(k9_openlatt(A),k9_openlatt(A)),k9_openlatt(A)) )
=> ( B = k13_openlatt(A)
<=> ! [C] :
( m1_subset_1(C,k9_openlatt(A))
=> ! [D] :
( m1_subset_1(D,k9_openlatt(A))
=> k2_binop_1(k9_openlatt(A),k9_openlatt(A),k9_openlatt(A),B,C,D) = k11_openlatt(A,C,D) ) ) ) ) ) ).
fof(d10_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(k9_openlatt(A),k9_openlatt(A)),k9_openlatt(A))
& m2_relset_1(B,k2_zfmisc_1(k9_openlatt(A),k9_openlatt(A)),k9_openlatt(A)) )
=> ( B = k14_openlatt(A)
<=> ! [C] :
( m1_subset_1(C,k9_openlatt(A))
=> ! [D] :
( m1_subset_1(D,k9_openlatt(A))
=> k2_binop_1(k9_openlatt(A),k9_openlatt(A),k9_openlatt(A),B,C,D) = k12_openlatt(A,C,D) ) ) ) ) ) ).
fof(t26_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> ( ~ v3_struct_0(g3_lattices(k9_openlatt(A),k13_openlatt(A),k14_openlatt(A)))
& v10_lattices(g3_lattices(k9_openlatt(A),k13_openlatt(A),k14_openlatt(A)))
& l3_lattices(g3_lattices(k9_openlatt(A),k13_openlatt(A),k14_openlatt(A))) ) ) ).
fof(d11_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> k15_openlatt(A) = g3_lattices(k9_openlatt(A),k13_openlatt(A),k14_openlatt(A)) ) ).
fof(t27_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> u1_struct_0(k15_openlatt(A)) = k9_openlatt(A) ) ).
fof(t28_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_openlatt(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_openlatt(A)))
=> ( k3_lattices(k15_openlatt(A),B,C) = k2_xboole_0(B,C)
& k4_lattices(k15_openlatt(A),B,C) = k3_xboole_0(B,C) ) ) ) ) ).
fof(t29_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_openlatt(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_openlatt(A)))
=> ( r3_lattices(k15_openlatt(A),B,C)
<=> r1_tarski(B,C) ) ) ) ) ).
fof(t30_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> v3_lattice4(k16_openlatt(A),A,k15_openlatt(A)) ) ).
fof(t31_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> v11_lattices(k15_openlatt(A)) ) ).
fof(t32_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> r1_filter_1(A,k15_openlatt(A)) ) ).
fof(t33_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v1_lattice2(A)
& ~ v3_realset2(A)
& l3_lattices(A) )
=> k8_funct_2(u1_struct_0(A),u1_struct_0(k15_openlatt(A)),k16_openlatt(A),k6_lattices(A)) = k7_openlatt(A) ) ).
fof(t34_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v1_lattice2(A)
& ~ v3_realset2(A)
& l3_lattices(A) )
=> k8_funct_2(u1_struct_0(A),u1_struct_0(k15_openlatt(A)),k16_openlatt(A),k5_lattices(A)) = k1_xboole_0 ) ).
fof(t35_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v1_lattice2(A)
& ~ v3_realset2(A)
& l3_lattices(A) )
=> r1_tarski(k9_openlatt(A),k1_zfmisc_1(k7_openlatt(A))) ) ).
fof(t37_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v1_lattice2(A)
& ~ v3_realset2(A)
& l3_lattices(A) )
=> r1_tarski(k9_openlatt(A),u1_struct_0(k6_openlatt(k17_openlatt(A)))) ) ).
fof(t38_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v1_lattice2(A)
& ~ v3_realset2(A)
& l3_lattices(A) )
=> v1_lattice4(k18_openlatt(A),A,k6_openlatt(k17_openlatt(A))) ) ).
fof(t39_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v1_lattice2(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))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(k6_openlatt(k17_openlatt(A))),k18_openlatt(A),k4_filter_0(A,B,C)) = k4_filter_0(k6_openlatt(k17_openlatt(A)),k8_funct_2(u1_struct_0(A),u1_struct_0(k6_openlatt(k17_openlatt(A))),k18_openlatt(A),B),k8_funct_2(u1_struct_0(A),u1_struct_0(k6_openlatt(k17_openlatt(A))),k18_openlatt(A),C)) ) ) ) ).
fof(t40_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v1_lattice2(A)
& ~ v3_realset2(A)
& l3_lattices(A) )
=> r1_lattice4(A,k6_openlatt(k17_openlatt(A)),k18_openlatt(A)) ) ).
fof(t41_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v1_lattice2(A)
& ~ v3_realset2(A)
& l3_lattices(A) )
=> r2_lattice4(A,k6_openlatt(k17_openlatt(A)),k18_openlatt(A)) ) ).
fof(t42_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v1_lattice2(A)
& ~ v3_realset2(A)
& l3_lattices(A) )
=> r3_lattice4(A,k6_openlatt(k17_openlatt(A)),k18_openlatt(A)) ) ).
fof(dt_k1_openlatt,axiom,
! [A] :
( l1_pre_topc(A)
=> m1_subset_1(k1_openlatt(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ).
fof(dt_k2_openlatt,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_openlatt(A))
& m1_subset_1(C,k1_openlatt(A)) )
=> m2_subset_1(k2_openlatt(A,B,C),k1_zfmisc_1(u1_struct_0(A)),k1_openlatt(A)) ) ).
fof(commutativity_k2_openlatt,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_openlatt(A))
& m1_subset_1(C,k1_openlatt(A)) )
=> k2_openlatt(A,B,C) = k2_openlatt(A,C,B) ) ).
fof(idempotence_k2_openlatt,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_openlatt(A))
& m1_subset_1(C,k1_openlatt(A)) )
=> k2_openlatt(A,B,B) = B ) ).
fof(redefinition_k2_openlatt,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_openlatt(A))
& m1_subset_1(C,k1_openlatt(A)) )
=> k2_openlatt(A,B,C) = k2_xboole_0(B,C) ) ).
fof(dt_k3_openlatt,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_openlatt(A))
& m1_subset_1(C,k1_openlatt(A)) )
=> m2_subset_1(k3_openlatt(A,B,C),k1_zfmisc_1(u1_struct_0(A)),k1_openlatt(A)) ) ).
fof(commutativity_k3_openlatt,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_openlatt(A))
& m1_subset_1(C,k1_openlatt(A)) )
=> k3_openlatt(A,B,C) = k3_openlatt(A,C,B) ) ).
fof(idempotence_k3_openlatt,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_openlatt(A))
& m1_subset_1(C,k1_openlatt(A)) )
=> k3_openlatt(A,B,B) = B ) ).
fof(redefinition_k3_openlatt,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_openlatt(A))
& m1_subset_1(C,k1_openlatt(A)) )
=> k3_openlatt(A,B,C) = k3_xboole_0(B,C) ) ).
fof(dt_k4_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_funct_1(k4_openlatt(A))
& v1_funct_2(k4_openlatt(A),k2_zfmisc_1(k1_openlatt(A),k1_openlatt(A)),k1_openlatt(A))
& m2_relset_1(k4_openlatt(A),k2_zfmisc_1(k1_openlatt(A),k1_openlatt(A)),k1_openlatt(A)) ) ) ).
fof(dt_k5_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_funct_1(k5_openlatt(A))
& v1_funct_2(k5_openlatt(A),k2_zfmisc_1(k1_openlatt(A),k1_openlatt(A)),k1_openlatt(A))
& m2_relset_1(k5_openlatt(A),k2_zfmisc_1(k1_openlatt(A),k1_openlatt(A)),k1_openlatt(A)) ) ) ).
fof(dt_k6_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ~ v3_struct_0(k6_openlatt(A))
& v10_lattices(k6_openlatt(A))
& l3_lattices(k6_openlatt(A)) ) ) ).
fof(dt_k7_openlatt,axiom,
$true ).
fof(dt_k8_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> ( v1_relat_1(k8_openlatt(A))
& v1_funct_1(k8_openlatt(A)) ) ) ).
fof(dt_k9_openlatt,axiom,
$true ).
fof(dt_k10_openlatt,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_subset_1(k10_openlatt(A,B),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ).
fof(dt_k11_openlatt,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A)
& m1_subset_1(B,k9_openlatt(A))
& m1_subset_1(C,k9_openlatt(A)) )
=> m1_subset_1(k11_openlatt(A,B,C),k9_openlatt(A)) ) ).
fof(commutativity_k11_openlatt,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A)
& m1_subset_1(B,k9_openlatt(A))
& m1_subset_1(C,k9_openlatt(A)) )
=> k11_openlatt(A,B,C) = k11_openlatt(A,C,B) ) ).
fof(idempotence_k11_openlatt,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A)
& m1_subset_1(B,k9_openlatt(A))
& m1_subset_1(C,k9_openlatt(A)) )
=> k11_openlatt(A,B,B) = B ) ).
fof(redefinition_k11_openlatt,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A)
& m1_subset_1(B,k9_openlatt(A))
& m1_subset_1(C,k9_openlatt(A)) )
=> k11_openlatt(A,B,C) = k2_xboole_0(B,C) ) ).
fof(dt_k12_openlatt,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A)
& m1_subset_1(B,k9_openlatt(A))
& m1_subset_1(C,k9_openlatt(A)) )
=> m1_subset_1(k12_openlatt(A,B,C),k9_openlatt(A)) ) ).
fof(commutativity_k12_openlatt,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A)
& m1_subset_1(B,k9_openlatt(A))
& m1_subset_1(C,k9_openlatt(A)) )
=> k12_openlatt(A,B,C) = k12_openlatt(A,C,B) ) ).
fof(idempotence_k12_openlatt,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A)
& m1_subset_1(B,k9_openlatt(A))
& m1_subset_1(C,k9_openlatt(A)) )
=> k12_openlatt(A,B,B) = B ) ).
fof(redefinition_k12_openlatt,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A)
& m1_subset_1(B,k9_openlatt(A))
& m1_subset_1(C,k9_openlatt(A)) )
=> k12_openlatt(A,B,C) = k3_xboole_0(B,C) ) ).
fof(dt_k13_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> ( v1_funct_1(k13_openlatt(A))
& v1_funct_2(k13_openlatt(A),k2_zfmisc_1(k9_openlatt(A),k9_openlatt(A)),k9_openlatt(A))
& m2_relset_1(k13_openlatt(A),k2_zfmisc_1(k9_openlatt(A),k9_openlatt(A)),k9_openlatt(A)) ) ) ).
fof(dt_k14_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> ( v1_funct_1(k14_openlatt(A))
& v1_funct_2(k14_openlatt(A),k2_zfmisc_1(k9_openlatt(A),k9_openlatt(A)),k9_openlatt(A))
& m2_relset_1(k14_openlatt(A),k2_zfmisc_1(k9_openlatt(A),k9_openlatt(A)),k9_openlatt(A)) ) ) ).
fof(dt_k15_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> ( ~ v3_struct_0(k15_openlatt(A))
& v10_lattices(k15_openlatt(A))
& l3_lattices(k15_openlatt(A)) ) ) ).
fof(dt_k16_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> m1_lattice4(k16_openlatt(A),A,k15_openlatt(A)) ) ).
fof(redefinition_k16_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> k16_openlatt(A) = k8_openlatt(A) ) ).
fof(dt_k17_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v1_lattice2(A)
& ~ v3_realset2(A)
& l3_lattices(A) )
=> ( v1_pre_topc(k17_openlatt(A))
& l1_pre_topc(k17_openlatt(A)) ) ) ).
fof(dt_k18_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v1_lattice2(A)
& ~ v3_realset2(A)
& l3_lattices(A) )
=> m1_lattice4(k18_openlatt(A),A,k6_openlatt(k17_openlatt(A))) ) ).
fof(redefinition_k18_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v1_lattice2(A)
& ~ v3_realset2(A)
& l3_lattices(A) )
=> k18_openlatt(A) = k8_openlatt(A) ) ).
fof(d5_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> k7_openlatt(A) = a_1_0_openlatt(A) ) ).
fof(d6_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( B = k8_openlatt(A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k1_relat_1(B) = u1_struct_0(A)
& k1_funct_1(B,C) = a_2_0_openlatt(A,C) ) ) ) ) ) ).
fof(d8_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k10_openlatt(A,B) = a_2_1_openlatt(A,B) ) ) ).
fof(d12_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v1_lattice2(A)
& ~ v3_realset2(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_pre_topc(B)
& l1_pre_topc(B) )
=> ( B = k17_openlatt(A)
<=> ( u1_struct_0(B) = k7_openlatt(A)
& u1_pre_topc(B) = a_1_1_openlatt(A) ) ) ) ) ).
fof(t36_openlatt,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v1_lattice2(A)
& ~ v3_realset2(A)
& l3_lattices(A) )
=> u1_struct_0(k6_openlatt(k17_openlatt(A))) = a_1_1_openlatt(A) ) ).
fof(fraenkel_a_1_0_openlatt,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& v11_lattices(B)
& l3_lattices(B) )
=> ( r2_hidden(A,a_1_0_openlatt(B))
<=> ? [C] :
( m1_filter_0(C,B)
& A = C
& C != u1_struct_0(B)
& v2_filter_0(C,B) ) ) ) ).
fof(fraenkel_a_2_0_openlatt,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& v11_lattices(B)
& l3_lattices(B)
& m1_subset_1(C,u1_struct_0(B)) )
=> ( r2_hidden(A,a_2_0_openlatt(B,C))
<=> ? [D] :
( m1_filter_0(D,B)
& A = D
& r2_hidden(D,k7_openlatt(B))
& r2_hidden(C,D) ) ) ) ).
fof(fraenkel_a_2_1_openlatt,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& v11_lattices(B)
& l3_lattices(B)
& m1_subset_1(C,u1_struct_0(B)) )
=> ( r2_hidden(A,a_2_1_openlatt(B,C))
<=> ? [D] :
( m1_filter_0(D,B)
& A = D
& r2_hidden(C,D) ) ) ) ).
fof(fraenkel_a_1_1_openlatt,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& v1_lattice2(B)
& ~ v3_realset2(B)
& l3_lattices(B) )
=> ( r2_hidden(A,a_1_1_openlatt(B))
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(k9_openlatt(B)))
& A = k3_tarski(C) ) ) ) ).
%------------------------------------------------------------------------------