SET007 Axioms: SET007+580.ax
%------------------------------------------------------------------------------
% File : SET007+580 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Lattice of Substitutions is a Heyting Algebra
% 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 : heyting2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 56 ( 2 unt; 0 def)
% Number of atoms : 290 ( 49 equ)
% Maximal formula atoms : 16 ( 5 avg)
% Number of connectives : 257 ( 23 ~; 0 |; 90 &)
% ( 10 <=>; 134 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 9 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 30 ( 28 usr; 1 prp; 0-3 aty)
% Number of functors : 44 ( 44 usr; 1 con; 0-6 aty)
% Number of variables : 216 ( 210 !; 6 ?)
% 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_heyting2,axiom,
! [A,B] :
( ~ v1_xboole_0(k1_tarski(k4_tarski(A,B)))
& v1_relat_1(k1_tarski(k4_tarski(A,B)))
& v1_funct_1(k1_tarski(k4_tarski(A,B))) ) ).
fof(rc1_heyting2,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_fraenkel(A) ) ).
fof(fc2_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ( ~ v3_struct_0(k5_substlat(A,B))
& v3_lattices(k5_substlat(A,B))
& v4_lattices(k5_substlat(A,B))
& v5_lattices(k5_substlat(A,B))
& v6_lattices(k5_substlat(A,B))
& v7_lattices(k5_substlat(A,B))
& v8_lattices(k5_substlat(A,B))
& v9_lattices(k5_substlat(A,B))
& v10_lattices(k5_substlat(A,B))
& v11_lattices(k5_substlat(A,B))
& v12_lattices(k5_substlat(A,B))
& v13_lattices(k5_substlat(A,B))
& v14_lattices(k5_substlat(A,B))
& v15_lattices(k5_substlat(A,B))
& v3_filter_0(k5_substlat(A,B)) ) ) ).
fof(t1_heyting2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ~ ! [C] :
( m1_subset_1(C,k4_partfun1(A,B))
=> C = k1_xboole_0 ) ) ) ).
fof(t2_heyting2,axiom,
! [A,B,C,D] :
( ( r2_hidden(D,k1_substlat(A,B))
& r2_hidden(C,D) )
=> ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finset_1(C) ) ) ).
fof(t3_heyting2,axiom,
! [A,B,C] :
( m1_subset_1(C,k4_partfun1(A,B))
=> ! [D] :
( r1_tarski(D,C)
=> r2_hidden(D,k4_partfun1(A,B)) ) ) ).
fof(t4_heyting2,axiom,
! [A,B] : r1_tarski(k4_partfun1(A,B),k1_zfmisc_1(k2_zfmisc_1(A,B))) ).
fof(t5_heyting2,axiom,
! [A,B] :
( ( v1_finset_1(A)
& v1_finset_1(B) )
=> v1_finset_1(k4_partfun1(A,B)) ) ).
fof(t6_heyting2,axiom,
! [A,B,C] :
( ( v1_finset_1(C)
& m1_subset_1(C,k4_partfun1(A,B)) )
=> r2_hidden(k2_setwiseo(k4_partfun1(A,B),C),k1_substlat(A,B)) ) ).
fof(t7_heyting2,axiom,
! [A,B,C] :
( m2_subset_1(C,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
=> ! [D] :
( m2_subset_1(D,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
=> ( k4_substlat(A,B,C,D) = C
=> ! [E] :
~ ( r2_hidden(E,C)
& ! [F] :
~ ( r2_hidden(F,D)
& r1_tarski(F,E) ) ) ) ) ) ).
fof(t8_heyting2,axiom,
! [A,B,C] :
( m2_subset_1(C,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
=> ! [D] :
( m2_subset_1(D,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
=> ( k3_substlat(A,B,k4_substlat(A,B,C,D)) = C
=> ! [E] :
~ ( r2_hidden(E,C)
& ! [F] :
~ ( r2_hidden(F,D)
& r1_tarski(F,E) ) ) ) ) ) ).
fof(t9_heyting2,axiom,
! [A,B,C] :
( m2_subset_1(C,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
=> ! [D] :
( m2_subset_1(D,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
=> ( ! [E] :
~ ( r2_hidden(E,C)
& ! [F] :
~ ( r2_hidden(F,D)
& r1_tarski(F,E) ) )
=> k3_substlat(A,B,k4_substlat(A,B,C,D)) = C ) ) ) ).
fof(d1_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> ! [D] :
( D = k1_heyting2(A,B,C)
<=> ! [E] :
( r2_hidden(E,D)
<=> ? [F] :
( v1_relat_1(F)
& v1_funct_1(F)
& v1_finset_1(F)
& r2_hidden(F,C)
& r2_hidden(E,k1_relat_1(F)) ) ) ) ) ) ).
fof(t10_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> r1_tarski(k1_heyting2(A,B,C),A) ) ) ).
fof(t11_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> ( C = k1_xboole_0
=> k1_heyting2(A,B,C) = k1_xboole_0 ) ) ) ).
fof(t12_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> v1_finset_1(k1_heyting2(A,B,C)) ) ) ).
fof(t13_heyting2,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m1_subset_1(B,k5_finsub_1(k4_partfun1(k1_xboole_0,A)))
=> k1_heyting2(k1_xboole_0,A,B) = k1_xboole_0 ) ) ).
fof(t14_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( m2_subset_1(C,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
=> k4_substlat(A,B,C,k2_heyting2(A,B,C)) = k1_xboole_0 ) ) ).
fof(t15_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( m2_subset_1(C,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
=> ( C = k1_xboole_0
=> k2_heyting2(A,B,C) = k1_tarski(k1_xboole_0) ) ) ) ).
fof(t16_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( m2_subset_1(C,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
=> ( C = k1_tarski(k1_xboole_0)
=> k2_heyting2(A,B,C) = k1_xboole_0 ) ) ) ).
fof(t17_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( m2_subset_1(C,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
=> k3_substlat(A,B,k4_substlat(A,B,C,k2_heyting2(A,B,C))) = k5_lattices(k5_substlat(A,B)) ) ) ).
fof(t18_heyting2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_finset_1(B) )
=> ! [C] :
( m2_subset_1(C,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
=> ( C = k1_xboole_0
=> k3_substlat(A,B,k2_heyting2(A,B,C)) = k6_lattices(k5_substlat(A,B)) ) ) ) ) ).
fof(t19_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( m2_subset_1(C,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
=> ! [D] :
( m1_subset_1(D,k4_partfun1(A,B))
=> ! [E] :
( m2_subset_1(E,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
=> ~ ( E = k2_setwiseo(k4_partfun1(A,B),D)
& k4_substlat(A,B,C,E) = k1_xboole_0
& ! [F] :
( v1_finset_1(F)
=> ~ ( r2_hidden(F,k2_heyting2(A,B,C))
& r1_tarski(F,D) ) ) ) ) ) ) ) ).
fof(t21_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> ( C = k1_xboole_0
=> k3_heyting2(A,B,C,C) = k1_tarski(k1_xboole_0) ) ) ) ).
fof(t22_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k5_substlat(A,B)))
=> ! [D] :
( r1_tarski(D,C)
=> m1_subset_1(D,u1_struct_0(k5_substlat(A,B))) ) ) ) ).
fof(d4_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)))
& m2_relset_1(C,u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B))) )
=> ( C = k4_heyting2(A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(k5_substlat(A,B)))
=> ! [E] :
( m2_subset_1(E,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
=> ( E = D
=> k8_funct_2(u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)),C,D) = k3_substlat(A,B,k2_heyting2(A,B,E)) ) ) ) ) ) ) ).
fof(d5_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B))),u1_struct_0(k5_substlat(A,B)))
& m2_relset_1(C,k2_zfmisc_1(u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B))),u1_struct_0(k5_substlat(A,B))) )
=> ( C = k5_heyting2(A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(k5_substlat(A,B)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k5_substlat(A,B)))
=> ! [F] :
( m2_subset_1(F,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
=> ! [G] :
( m2_subset_1(G,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
=> ( ( F = D
& G = E )
=> k2_binop_1(u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)),C,D,E) = k3_substlat(A,B,k3_heyting2(A,B,F,G)) ) ) ) ) ) ) ) ) ).
fof(d6_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k5_substlat(A,B)))
=> k6_heyting2(A,B,C) = k1_zfmisc_1(C) ) ) ).
fof(d7_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k5_substlat(A,B)))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)))
& m2_relset_1(D,u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B))) )
=> ( D = k7_heyting2(A,B,C)
<=> ! [E] :
( m1_subset_1(E,u1_struct_0(k5_substlat(A,B)))
=> k8_funct_2(u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)),D,E) = k4_xboole_0(C,E) ) ) ) ) ) ).
fof(d8_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k4_partfun1(A,B),u1_struct_0(k5_substlat(A,B)))
& m2_relset_1(C,k4_partfun1(A,B),u1_struct_0(k5_substlat(A,B))) )
=> ( C = k8_heyting2(A,B)
<=> ! [D] :
( m1_subset_1(D,k4_partfun1(A,B))
=> k8_funct_2(k4_partfun1(A,B),u1_struct_0(k5_substlat(A,B)),C,D) = k3_substlat(A,B,k2_setwiseo(k4_partfun1(A,B),D)) ) ) ) ) ).
fof(t23_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( m2_subset_1(C,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
=> k2_lattice2(k4_partfun1(A,B),k5_substlat(A,B),C,k8_heyting2(A,B)) = k10_setwiseo(k4_partfun1(A,B),k4_partfun1(A,B),C,k11_setwiseo(k4_partfun1(A,B))) ) ) ).
fof(t24_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( m2_subset_1(C,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
=> C = k2_lattice2(k4_partfun1(A,B),k5_substlat(A,B),C,k8_heyting2(A,B)) ) ) ).
fof(t25_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k5_substlat(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k5_substlat(A,B)))
=> r3_lattices(k5_substlat(A,B),k8_funct_2(u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)),k7_heyting2(A,B,C),D),C) ) ) ) ).
fof(t26_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( m1_subset_1(C,k4_partfun1(A,B))
=> ( v1_finset_1(C)
=> ! [D] :
( r2_hidden(D,k8_funct_2(k4_partfun1(A,B),u1_struct_0(k5_substlat(A,B)),k8_heyting2(A,B),C))
=> D = C ) ) ) ) ).
fof(t27_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k5_substlat(A,B)))
=> ! [D] :
( m2_subset_1(D,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
=> ! [E] :
( m2_subset_1(E,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
=> ! [F] :
( m1_subset_1(F,k4_partfun1(A,B))
=> ( ( D = k2_setwiseo(k4_partfun1(A,B),F)
& E = C
& k4_substlat(A,B,E,D) = k1_xboole_0 )
=> r3_lattices(k5_substlat(A,B),k8_funct_2(k4_partfun1(A,B),u1_struct_0(k5_substlat(A,B)),k8_heyting2(A,B),F),k8_funct_2(u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)),k4_heyting2(A,B),C)) ) ) ) ) ) ) ).
fof(t28_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( ( v1_finset_1(C)
& m1_subset_1(C,k4_partfun1(A,B)) )
=> r2_hidden(C,k8_funct_2(k4_partfun1(A,B),u1_struct_0(k5_substlat(A,B)),k8_heyting2(A,B),C)) ) ) ).
fof(t29_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( m1_subset_1(C,k4_partfun1(A,B))
=> ! [D] :
( m2_subset_1(D,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
=> ! [E] :
( m2_subset_1(E,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
=> ~ ( ! [F] :
~ ( r2_hidden(F,D)
& ! [G] :
~ ( r2_hidden(G,E)
& r1_tarski(G,k2_xboole_0(F,C)) ) )
& ! [F] :
~ ( r2_hidden(F,k3_heyting2(A,B,D,E))
& r1_tarski(F,C) ) ) ) ) ) ) ).
fof(t30_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k5_substlat(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k5_substlat(A,B)))
=> ! [E] :
( ( v1_finset_1(E)
& m1_subset_1(E,k4_partfun1(A,B)) )
=> ( ( ! [F] :
( m1_subset_1(F,k4_partfun1(A,B))
=> ( r2_hidden(F,C)
=> r1_partfun1(F,E) ) )
& r3_lattices(k5_substlat(A,B),k4_lattices(k5_substlat(A,B),C,k8_funct_2(k4_partfun1(A,B),u1_struct_0(k5_substlat(A,B)),k8_heyting2(A,B),E)),D) )
=> r3_lattices(k5_substlat(A,B),k8_funct_2(k4_partfun1(A,B),u1_struct_0(k5_substlat(A,B)),k8_heyting2(A,B),E),k2_binop_1(u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)),k5_heyting2(A,B),C,D)) ) ) ) ) ) ).
fof(t31_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k5_substlat(A,B)))
=> k4_lattices(k5_substlat(A,B),C,k8_funct_2(u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)),k4_heyting2(A,B),C)) = k5_lattices(k5_substlat(A,B)) ) ) ).
fof(t32_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k5_substlat(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k5_substlat(A,B)))
=> r3_lattices(k5_substlat(A,B),k4_lattices(k5_substlat(A,B),C,k2_binop_1(u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)),k5_heyting2(A,B),C,D)),D) ) ) ) ).
fof(t33_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k5_substlat(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k5_substlat(A,B)))
=> k4_filter_0(k5_substlat(A,B),C,D) = k2_lattice2(u1_struct_0(k5_substlat(A,B)),k5_substlat(A,B),k6_heyting2(A,B,C),k6_funcop_1(u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)),u1_lattices(k5_substlat(A,B)),k4_heyting2(A,B),k7_funcop_1(u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)),k5_heyting2(A,B),k7_heyting2(A,B,C),D))) ) ) ) ).
fof(dt_k1_heyting2,axiom,
$true ).
fof(dt_k2_heyting2,axiom,
! [A,B,C] :
( ( v1_finset_1(B)
& m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B))) )
=> m1_subset_1(k2_heyting2(A,B,C),k5_finsub_1(k4_partfun1(A,B))) ) ).
fof(dt_k3_heyting2,axiom,
! [A,B,C,D] :
( ( v1_finset_1(B)
& m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
& m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B))) )
=> m1_subset_1(k3_heyting2(A,B,C,D),k5_finsub_1(k4_partfun1(A,B))) ) ).
fof(dt_k4_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ( v1_funct_1(k4_heyting2(A,B))
& v1_funct_2(k4_heyting2(A,B),u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)))
& m2_relset_1(k4_heyting2(A,B),u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B))) ) ) ).
fof(dt_k5_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ( v1_funct_1(k5_heyting2(A,B))
& v1_funct_2(k5_heyting2(A,B),k2_zfmisc_1(u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B))),u1_struct_0(k5_substlat(A,B)))
& m2_relset_1(k5_heyting2(A,B),k2_zfmisc_1(u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B))),u1_struct_0(k5_substlat(A,B))) ) ) ).
fof(dt_k6_heyting2,axiom,
! [A,B,C] :
( ( v1_finset_1(B)
& m1_subset_1(C,u1_struct_0(k5_substlat(A,B))) )
=> m1_subset_1(k6_heyting2(A,B,C),k5_finsub_1(u1_struct_0(k5_substlat(A,B)))) ) ).
fof(dt_k7_heyting2,axiom,
! [A,B,C] :
( ( v1_finset_1(B)
& m1_subset_1(C,u1_struct_0(k5_substlat(A,B))) )
=> ( v1_funct_1(k7_heyting2(A,B,C))
& v1_funct_2(k7_heyting2(A,B,C),u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)))
& m2_relset_1(k7_heyting2(A,B,C),u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B))) ) ) ).
fof(dt_k8_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ( v1_funct_1(k8_heyting2(A,B))
& v1_funct_2(k8_heyting2(A,B),k4_partfun1(A,B),u1_struct_0(k5_substlat(A,B)))
& m2_relset_1(k8_heyting2(A,B),k4_partfun1(A,B),u1_struct_0(k5_substlat(A,B))) ) ) ).
fof(d2_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> k2_heyting2(A,B,C) = a_3_0_heyting2(A,B,C) ) ) ).
fof(d3_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
=> k3_heyting2(A,B,C,D) = k3_xboole_0(k4_partfun1(A,B),a_4_0_heyting2(A,B,C,D)) ) ) ) ).
fof(t20_heyting2,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
=> ! [E] :
~ ( r2_hidden(E,k3_heyting2(A,B,C,D))
& ! [F] :
( ( v1_funct_1(F)
& m2_relset_1(F,C,D) )
=> ~ ( E = k3_tarski(a_5_1_heyting2(A,B,C,D,F))
& k4_relset_1(C,D,F) = C ) ) ) ) ) ) ).
fof(fraenkel_a_3_0_heyting2,axiom,
! [A,B,C,D] :
( ( v1_finset_1(C)
& m1_subset_1(D,k5_finsub_1(k4_partfun1(B,C))) )
=> ( r2_hidden(A,a_3_0_heyting2(B,C,D))
<=> ? [E] :
( m1_subset_1(E,k4_partfun1(k1_heyting2(B,C,D),C))
& A = E
& ! [F] :
( m1_subset_1(F,k4_partfun1(B,C))
=> ~ ( r2_hidden(F,D)
& r1_partfun1(E,F) ) ) ) ) ) ).
fof(fraenkel_a_4_0_heyting2,axiom,
! [A,B,C,D,E] :
( ( v1_finset_1(C)
& m1_subset_1(D,k5_finsub_1(k4_partfun1(B,C)))
& m1_subset_1(E,k5_finsub_1(k4_partfun1(B,C))) )
=> ( r2_hidden(A,a_4_0_heyting2(B,C,D,E))
<=> ? [F] :
( m1_subset_1(F,k4_partfun1(D,E))
& A = k3_tarski(a_5_0_heyting2(B,C,D,E,F))
& k1_relat_1(F) = D ) ) ) ).
fof(fraenkel_a_5_0_heyting2,axiom,
! [A,B,C,D,E,F] :
( ( v1_finset_1(C)
& m1_subset_1(D,k5_finsub_1(k4_partfun1(B,C)))
& m1_subset_1(E,k5_finsub_1(k4_partfun1(B,C)))
& m1_subset_1(F,k4_partfun1(D,E)) )
=> ( r2_hidden(A,a_5_0_heyting2(B,C,D,E,F))
<=> ? [G] :
( m1_subset_1(G,k4_partfun1(B,C))
& A = k4_xboole_0(k1_funct_1(F,G),G)
& r2_hidden(G,D) ) ) ) ).
fof(fraenkel_a_5_1_heyting2,axiom,
! [A,B,C,D,E,F] :
( ( v1_finset_1(C)
& m1_subset_1(D,k5_finsub_1(k4_partfun1(B,C)))
& m1_subset_1(E,k5_finsub_1(k4_partfun1(B,C)))
& v1_funct_1(F)
& m2_relset_1(F,D,E) )
=> ( r2_hidden(A,a_5_1_heyting2(B,C,D,E,F))
<=> ? [G] :
( m1_subset_1(G,k4_partfun1(B,C))
& A = k4_xboole_0(k1_funct_1(F,G),G)
& r2_hidden(G,D) ) ) ) ).
%------------------------------------------------------------------------------