SET007 Axioms: SET007+428.ax
%------------------------------------------------------------------------------
% File : SET007+428 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Extensions of Mappings on Generator Set
% 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 : extens_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 28 ( 3 unt; 0 def)
% Number of atoms : 209 ( 6 equ)
% Maximal formula atoms : 18 ( 7 avg)
% Number of connectives : 198 ( 17 ~; 0 |; 61 &)
% ( 4 <=>; 116 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 10 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 27 ( 25 usr; 1 prp; 0-4 aty)
% Number of functors : 16 ( 16 usr; 0 con; 1-6 aty)
% Number of variables : 118 ( 117 !; 1 ?)
% 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(rc1_extens_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ? [C] :
( m1_msafree(C,A,B)
& v1_relat_1(C)
& v2_relat_1(C)
& ~ v3_relat_1(C)
& v1_funct_1(C) ) ) ).
fof(t1_extens_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( r1_tarski(B,C)
=> k9_relat_1(k7_relat_1(A,C),B) = k9_relat_1(A,B) ) ) ).
fof(t2_extens_1,axiom,
$true ).
fof(t3_extens_1,axiom,
$true ).
fof(t4_extens_1,axiom,
$true ).
fof(t5_extens_1,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,A) )
=> ! [D] :
( m3_pboole(D,A,B,C)
=> ! [E] :
( m4_pboole(E,A,B)
=> ( r2_pboole(A,B,E)
=> r6_pboole(A,k1_msafree(A,B,C,E,D),D) ) ) ) ) ) ).
fof(t6_extens_1,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m4_pboole(D,A,B)
=> ! [E] :
( m3_pboole(E,A,B,C)
=> r2_pboole(A,k14_pboole(A,D,E),k14_pboole(A,B,E)) ) ) ) ) ).
fof(t7_extens_1,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,A) )
=> ! [D] :
( m3_pboole(D,A,B,C)
=> ! [E] :
( m4_pboole(E,A,B)
=> ! [F] :
( m4_pboole(F,A,B)
=> ( r2_pboole(A,E,F)
=> r6_pboole(A,k14_pboole(A,E,k1_msafree(A,B,C,F,D)),k14_pboole(A,E,D)) ) ) ) ) ) ) ).
fof(t8_extens_1,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,A) )
=> ! [D] :
( ( v2_relat_1(D)
& m1_pboole(D,A) )
=> ! [E] :
( m3_pboole(E,A,B,C)
=> ! [F] :
( m3_pboole(F,A,C,D)
=> ! [G] :
( m4_pboole(G,A,B)
=> r6_pboole(A,k1_msafree(A,B,D,G,k3_msualg_3(A,B,C,D,E,F)),k3_msualg_3(A,G,C,D,k1_msafree(A,B,C,G,E),F)) ) ) ) ) ) ) ).
fof(t9_extens_1,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ( r1_pzfmisc1(A,B,C)
=> ! [D] :
( m3_pboole(D,A,B,C)
=> ! [E] :
( m1_pboole(E,A)
=> ( m4_pboole(C,A,E)
=> m3_pboole(D,A,B,E) ) ) ) ) ) ) ).
fof(t10_extens_1,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,A) )
=> ! [D] :
( m3_pboole(D,A,B,C)
=> ! [E] :
( m4_pboole(E,A,B)
=> ( v1_msualg_3(D)
=> v1_msualg_3(k1_msafree(A,B,C,E,D)) ) ) ) ) ) ).
fof(t11_extens_1,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,A) )
=> ! [D] :
( m3_pboole(D,A,B,C)
=> ! [E] :
( m4_pboole(E,A,B)
=> r2_pboole(A,k1_extens_1(A,k1_msafree(A,B,C,E,D)),k1_extens_1(A,D)) ) ) ) ) ).
fof(t12_extens_1,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,A) )
=> ! [D] :
( m3_pboole(D,A,B,C)
=> ! [E] :
( m4_pboole(E,A,B)
=> r2_pboole(A,k2_extens_1(A,k1_msafree(A,B,C,E,D)),k2_extens_1(A,D)) ) ) ) ) ).
fof(t13_extens_1,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m3_pboole(D,A,B,C)
=> ( v2_msualg_3(D,A,B,C)
<=> r6_pboole(A,k2_extens_1(A,D),C) ) ) ) ) ).
fof(t14_extens_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v2_relat_1(B)
& m1_pboole(B,u1_struct_0(A)) )
=> r6_pboole(u1_struct_0(A),k2_extens_1(u1_struct_0(A),k15_msafree(A,B)),B) ) ) ).
fof(t15_extens_1,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,A) )
=> ! [D] :
( ( v2_relat_1(D)
& m1_pboole(D,A) )
=> ! [E] :
( m3_pboole(E,A,B,C)
=> ! [F] :
( m3_pboole(F,A,C,D)
=> ! [G] :
( ( v2_relat_1(G)
& m4_pboole(G,A,C) )
=> ( r2_pboole(A,k2_extens_1(A,E),G)
=> k13_pboole(E,k1_msafree(A,C,D,G,F)) = k3_msualg_3(A,B,C,D,E,F) ) ) ) ) ) ) ) ).
fof(t16_extens_1,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,A) )
=> ! [D] :
( m3_pboole(D,A,B,C)
=> ( v2_msualg_3(D,A,B,C)
<=> ! [E] :
( ( v2_relat_1(E)
& m1_pboole(E,A) )
=> ! [F] :
( m3_pboole(F,A,C,E)
=> ! [G] :
( m3_pboole(G,A,C,E)
=> ( r6_pboole(A,k3_msualg_3(A,B,C,E,D,F),k3_msualg_3(A,B,C,E,D,G))
=> r6_pboole(A,F,G) ) ) ) ) ) ) ) ) ).
fof(t17_extens_1,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,A) )
=> ! [D] :
( m3_pboole(D,A,B,C)
=> ( ( v2_relat_1(B)
& v2_relat_1(C) )
=> ( v1_msualg_3(D)
<=> ! [E] :
( m1_pboole(E,A)
=> ! [F] :
( m3_pboole(F,A,E,B)
=> ! [G] :
( m3_pboole(G,A,E,B)
=> ( k13_pboole(F,D) = k13_pboole(G,D)
=> r6_pboole(A,F,G) ) ) ) ) ) ) ) ) ) ).
fof(t18_extens_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,u1_struct_0(A)) )
=> ! [D] :
( m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,k11_msafree(A,C)),u4_msualg_1(A,B))
=> ! [E] :
( m3_pboole(E,u1_struct_0(A),u4_msualg_1(A,k11_msafree(A,C)),u4_msualg_1(A,B))
=> ( ( r1_msualg_3(A,k11_msafree(A,C),B,D)
& r1_msualg_3(A,k11_msafree(A,C),B,E)
& r6_pboole(u1_struct_0(A),k1_msafree(u1_struct_0(A),u4_msualg_1(A,k11_msafree(A,C)),u4_msualg_1(A,B),k13_msafree(A,C),D),k1_msafree(u1_struct_0(A),u4_msualg_1(A,k11_msafree(A,C)),u4_msualg_1(A,B),k13_msafree(A,C),E)) )
=> r6_pboole(u1_struct_0(A),D,E) ) ) ) ) ) ) ).
fof(t19_extens_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( ( v5_msualg_1(C,A)
& l3_msualg_1(C,A) )
=> ! [D] :
( m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C))
=> ( ( r1_msualg_3(A,B,C,D)
& r2_msualg_3(A,B,C,D) )
=> ! [E] :
( ( v5_msualg_1(E,A)
& l3_msualg_1(E,A) )
=> ! [F] :
( m3_pboole(F,u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,E))
=> ! [G] :
( m3_pboole(G,u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,E))
=> ( ( r1_msualg_3(A,C,E,F)
& r1_msualg_3(A,C,E,G)
& r6_pboole(u1_struct_0(A),k3_msualg_3(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C),u4_msualg_1(A,E),D,F),k3_msualg_3(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C),u4_msualg_1(A,E),D,G)) )
=> r6_pboole(u1_struct_0(A),F,G) ) ) ) ) ) ) ) ) ) ).
fof(t20_extens_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( ( v5_msualg_1(C,A)
& l3_msualg_1(C,A) )
=> ! [D] :
( m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C))
=> ( r1_msualg_3(A,B,C,D)
=> ( r3_msualg_3(A,B,C,D)
<=> ! [E] :
( ( v5_msualg_1(E,A)
& l3_msualg_1(E,A) )
=> ! [F] :
( m3_pboole(F,u1_struct_0(A),u4_msualg_1(A,E),u4_msualg_1(A,B))
=> ! [G] :
( m3_pboole(G,u1_struct_0(A),u4_msualg_1(A,E),u4_msualg_1(A,B))
=> ( ( r1_msualg_3(A,E,B,F)
& r1_msualg_3(A,E,B,G)
& r6_pboole(u1_struct_0(A),k3_msualg_3(u1_struct_0(A),u4_msualg_1(A,E),u4_msualg_1(A,B),u4_msualg_1(A,C),F,D),k3_msualg_3(u1_struct_0(A),u4_msualg_1(A,E),u4_msualg_1(A,B),u4_msualg_1(A,C),G,D)) )
=> r6_pboole(u1_struct_0(A),F,G) ) ) ) ) ) ) ) ) ) ) ).
fof(t21_extens_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( l3_msualg_1(B,A)
=> ! [C] :
( m4_pboole(C,u1_struct_0(A),u4_msualg_1(A,B))
=> ! [D] :
( m4_pboole(D,u1_struct_0(A),u4_msualg_1(A,B))
=> ( m4_pboole(C,u1_struct_0(A),D)
=> m1_msualg_2(k12_msualg_2(A,B,C),A,k12_msualg_2(A,B,D)) ) ) ) ) ) ).
fof(t22_extens_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( l3_msualg_1(B,A)
=> ! [C] :
( m1_msualg_2(C,A,B)
=> ! [D] :
( m4_pboole(D,u1_struct_0(A),u4_msualg_1(A,B))
=> ! [E] :
( m4_pboole(E,u1_struct_0(A),u4_msualg_1(A,C))
=> ( r6_pboole(u1_struct_0(A),D,E)
=> k12_msualg_2(A,B,D) = k12_msualg_2(A,C,E) ) ) ) ) ) ) ).
fof(t23_extens_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v4_msualg_1(B,A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( ( v5_msualg_1(C,A)
& l3_msualg_1(C,A) )
=> ! [D] :
( m1_msafree(D,A,B)
=> ! [E] :
( m3_pboole(E,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C))
=> ! [F] :
( m3_pboole(F,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C))
=> ( ( r1_msualg_3(A,B,C,E)
& r1_msualg_3(A,B,C,F)
& r6_pboole(u1_struct_0(A),k1_msafree(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C),D,E),k1_msafree(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C),D,F)) )
=> r6_pboole(u1_struct_0(A),E,F) ) ) ) ) ) ) ) ).
fof(dt_k1_extens_1,axiom,
! [A,B] :
( ( v1_funcop_1(B)
& m1_pboole(B,A) )
=> m1_pboole(k1_extens_1(A,B),A) ) ).
fof(redefinition_k1_extens_1,axiom,
! [A,B] :
( ( v1_funcop_1(B)
& m1_pboole(B,A) )
=> k1_extens_1(A,B) = k2_funct_6(B) ) ).
fof(dt_k2_extens_1,axiom,
! [A,B] :
( ( v1_funcop_1(B)
& m1_pboole(B,A) )
=> m1_pboole(k2_extens_1(A,B),A) ) ).
fof(redefinition_k2_extens_1,axiom,
! [A,B] :
( ( v1_funcop_1(B)
& m1_pboole(B,A) )
=> k2_extens_1(A,B) = k3_funct_6(B) ) ).
%------------------------------------------------------------------------------