SET007 Axioms: SET007+457.ax
%------------------------------------------------------------------------------
% File : SET007+457 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : More on the Lattice of Many Sorted Equivalence Relations
% 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 : msualg_7 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 35 ( 0 unt; 0 def)
% Number of atoms : 274 ( 24 equ)
% Maximal formula atoms : 15 ( 7 avg)
% Number of connectives : 277 ( 38 ~; 2 |; 116 &)
% ( 8 <=>; 113 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 9 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 35 ( 34 usr; 0 prp; 1-4 aty)
% Number of functors : 25 ( 25 usr; 5 con; 0-3 aty)
% Number of variables : 104 ( 96 !; 8 ?)
% 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_msualg_7,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_pboole(B,A) )
=> ( ~ v3_struct_0(k5_msualg_5(A,B))
& v3_lattices(k5_msualg_5(A,B))
& v4_lattices(k5_msualg_5(A,B))
& v5_lattices(k5_msualg_5(A,B))
& v6_lattices(k5_msualg_5(A,B))
& v7_lattices(k5_msualg_5(A,B))
& v8_lattices(k5_msualg_5(A,B))
& v9_lattices(k5_msualg_5(A,B))
& v10_lattices(k5_msualg_5(A,B))
& v13_lattices(k5_msualg_5(A,B))
& v14_lattices(k5_msualg_5(A,B))
& v15_lattices(k5_msualg_5(A,B)) ) ) ).
fof(fc2_msualg_7,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_pboole(B,A) )
=> ( ~ v3_struct_0(k5_msualg_5(A,B))
& v3_lattices(k5_msualg_5(A,B))
& v4_lattices(k5_msualg_5(A,B))
& v5_lattices(k5_msualg_5(A,B))
& v6_lattices(k5_msualg_5(A,B))
& v7_lattices(k5_msualg_5(A,B))
& v8_lattices(k5_msualg_5(A,B))
& v9_lattices(k5_msualg_5(A,B))
& v10_lattices(k5_msualg_5(A,B))
& v13_lattices(k5_msualg_5(A,B))
& v14_lattices(k5_msualg_5(A,B))
& v15_lattices(k5_msualg_5(A,B))
& v4_lattice3(k5_msualg_5(A,B)) ) ) ).
fof(rc1_msualg_7,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ? [B] :
( m2_nat_lat(B,A)
& ~ v3_struct_0(B)
& v4_lattices(B)
& v5_lattices(B)
& v6_lattices(B)
& v7_lattices(B)
& v8_lattices(B)
& v9_lattices(B)
& v10_lattices(B)
& v4_lattice3(B) ) ) ).
fof(cc1_msualg_7,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( m2_nat_lat(B,A)
=> ( v1_msualg_7(B,A)
=> v4_lattice3(B) ) ) ) ).
fof(cc2_msualg_7,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( m2_nat_lat(B,A)
=> ( v2_msualg_7(B,A)
=> v4_lattice3(B) ) ) ) ).
fof(t1_msualg_7,axiom,
! [A,B] :
( r2_hidden(A,u1_struct_0(k2_msualg_5(B)))
<=> ( v3_relat_2(A)
& v8_relat_2(A)
& v1_partfun1(A,B,B)
& m2_relset_1(A,B,B) ) ) ).
fof(t2_msualg_7,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ( v2_msualg_4(k2_msualg_3(A,B),A,B)
& m1_msualg_4(k2_msualg_3(A,B),A,B,B) ) ) ) ).
fof(t3_msualg_7,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ( v2_msualg_4(k11_pboole(A,B,B),A,B)
& m1_msualg_4(k11_pboole(A,B,B),A,B,B) ) ) ) ).
fof(t4_msualg_7,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> k5_lattices(k5_msualg_5(A,B)) = k2_msualg_3(A,B) ) ) ).
fof(t5_msualg_7,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> k6_lattices(k5_msualg_5(A,B)) = k11_pboole(A,B,B) ) ) ).
fof(t6_msualg_7,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_msualg_5(A,B))))
=> m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,k11_pboole(A,B,B)))) ) ) ) ).
fof(t7_msualg_7,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k5_msualg_5(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k5_msualg_5(A,B)))
=> ! [E] :
( ( v2_msualg_4(E,A,B)
& m1_msualg_4(E,A,B,B) )
=> ! [F] :
( ( v2_msualg_4(F,A,B)
& m1_msualg_4(F,A,B,B) )
=> ( ( C = E
& D = F )
=> ( r3_lattices(k5_msualg_5(A,B),C,D)
<=> r2_pboole(A,E,F) ) ) ) ) ) ) ) ) ).
fof(t8_msualg_7,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_msualg_5(A,B))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,k11_pboole(A,B,B))))
=> ( D = C
=> ! [E] :
( ( v2_msualg_4(E,A,B)
& m1_msualg_4(E,A,B,B) )
=> ! [F] :
( ( v2_msualg_4(F,A,B)
& m1_msualg_4(F,A,B,B) )
=> ( ( r6_pboole(A,E,k6_mssubfam(A,k11_pboole(A,B,B),k5_closure2(A,k11_pboole(A,B,B),D)))
& r2_hidden(F,C) )
=> r2_pboole(A,E,F) ) ) ) ) ) ) ) ) ).
fof(t9_msualg_7,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_msualg_5(A,B))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,k11_pboole(A,B,B))))
=> ( D = C
=> ( v1_xboole_0(C)
| ( v2_msualg_4(k6_mssubfam(A,k11_pboole(A,B,B),k5_closure2(A,k11_pboole(A,B,B),D)),A,B)
& m1_msualg_4(k6_mssubfam(A,k11_pboole(A,B,B),k5_closure2(A,k11_pboole(A,B,B),D)),A,B,B) ) ) ) ) ) ) ) ).
fof(d1_msualg_7,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> ( v4_lattice3(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& r4_lattice3(A,C,B)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r4_lattice3(A,D,B)
=> r1_lattices(A,C,D) ) ) ) ) ) ) ).
fof(t10_msualg_7,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> v4_lattice3(k5_msualg_5(A,B)) ) ) ).
fof(t11_msualg_7,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_msualg_5(A,B))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,k11_pboole(A,B,B))))
=> ( D = C
=> ( v1_xboole_0(C)
| ! [E] :
( ( v2_msualg_4(E,A,B)
& m1_msualg_4(E,A,B,B) )
=> ! [F] :
( ( v2_msualg_4(F,A,B)
& m1_msualg_4(F,A,B,B) )
=> ( ( r6_pboole(A,E,k6_mssubfam(A,k11_pboole(A,B,B),k5_closure2(A,k11_pboole(A,B,B),D)))
& F = k16_lattice3(k5_msualg_5(A,B),C) )
=> r6_pboole(A,E,F) ) ) ) ) ) ) ) ) ) ).
fof(d2_msualg_7,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m2_nat_lat(B,A)
=> ( v1_msualg_7(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
=> r2_hidden(k16_lattice3(A,C),u1_struct_0(B)) ) ) ) ) ).
fof(d3_msualg_7,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m2_nat_lat(B,A)
=> ( v2_msualg_7(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
=> r2_hidden(k15_lattice3(A,C),u1_struct_0(B)) ) ) ) ) ).
fof(t12_msualg_7,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m2_nat_lat(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ( ( C = E
& D = F )
=> ( k3_lattices(A,C,D) = k3_lattices(B,E,F)
& k4_lattices(A,C,D) = k4_lattices(B,E,F) ) ) ) ) ) ) ) ) ).
fof(t13_msualg_7,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m2_nat_lat(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ( D = E
=> ( r3_lattice3(A,D,C)
<=> r3_lattice3(B,E,C) ) ) ) ) ) ) ) ).
fof(t14_msualg_7,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m2_nat_lat(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ( D = E
=> ( r4_lattice3(A,D,C)
<=> r4_lattice3(B,E,C) ) ) ) ) ) ) ) ).
fof(t15_msualg_7,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( m2_nat_lat(B,A)
=> ( v1_msualg_7(B,A)
=> v4_lattice3(B) ) ) ) ).
fof(t16_msualg_7,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( m2_nat_lat(B,A)
=> ( v2_msualg_7(B,A)
=> v4_lattice3(B) ) ) ) ).
fof(t17_msualg_7,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( m2_nat_lat(B,A)
=> ( v1_msualg_7(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
=> k16_lattice3(A,C) = k16_lattice3(B,C) ) ) ) ) ).
fof(t18_msualg_7,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( m2_nat_lat(B,A)
=> ( v2_msualg_7(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
=> k15_lattice3(A,C) = k15_lattice3(B,C) ) ) ) ) ).
fof(t19_msualg_7,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( m2_nat_lat(B,A)
=> ( v1_msualg_7(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ( C = D
=> k16_lattice3(A,C) = k16_lattice3(B,D) ) ) ) ) ) ) ).
fof(t20_msualg_7,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( m2_nat_lat(B,A)
=> ( v2_msualg_7(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ( C = D
=> k15_lattice3(A,C) = k15_lattice3(B,D) ) ) ) ) ) ) ).
fof(d4_msualg_7,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r1_xreal_0(A,B)
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_lattices(C)
& v10_lattices(C)
& l3_lattices(C) )
=> ( C = k1_msualg_7(A,B)
<=> ( u1_struct_0(C) = k1_rcomp_1(A,B)
& u2_lattices(C) = k1_realset1(k2_real_lat,k1_rcomp_1(A,B))
& u1_lattices(C) = k1_realset1(k1_real_lat,k1_rcomp_1(A,B)) ) ) ) ) ) ) ).
fof(t21_msualg_7,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r1_xreal_0(A,B)
=> v4_lattice3(k1_msualg_7(A,B)) ) ) ) ).
fof(t22_msualg_7,axiom,
? [A] :
( m2_nat_lat(A,k1_msualg_7(np__0,np__1))
& v2_msualg_7(A,k1_msualg_7(np__0,np__1))
& ~ v1_msualg_7(A,k1_msualg_7(np__0,np__1)) ) ).
fof(t23_msualg_7,axiom,
? [A] :
( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A)
& ? [B] :
( m2_nat_lat(B,A)
& v2_msualg_7(B,A)
& ~ v1_msualg_7(B,A) ) ) ).
fof(t24_msualg_7,axiom,
? [A] :
( m2_nat_lat(A,k1_msualg_7(np__0,np__1))
& v1_msualg_7(A,k1_msualg_7(np__0,np__1))
& ~ v2_msualg_7(A,k1_msualg_7(np__0,np__1)) ) ).
fof(t25_msualg_7,axiom,
? [A] :
( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A)
& ? [B] :
( m2_nat_lat(B,A)
& v1_msualg_7(B,A)
& ~ v2_msualg_7(B,A) ) ) ).
fof(dt_k1_msualg_7,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> ( ~ v3_struct_0(k1_msualg_7(A,B))
& v3_lattices(k1_msualg_7(A,B))
& v10_lattices(k1_msualg_7(A,B))
& l3_lattices(k1_msualg_7(A,B)) ) ) ).
%------------------------------------------------------------------------------