SET007 Axioms: SET007+654.ax
%------------------------------------------------------------------------------
% File : SET007+654 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Tichonov Theorem
% 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 : yellow17 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 26 ( 1 unt; 0 def)
% Number of atoms : 242 ( 18 equ)
% Maximal formula atoms : 18 ( 9 avg)
% Number of connectives : 259 ( 43 ~; 4 |; 93 &)
% ( 7 <=>; 112 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 12 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 22 ( 20 usr; 1 prp; 0-3 aty)
% Number of functors : 23 ( 23 usr; 3 con; 0-4 aty)
% Number of variables : 122 ( 118 !; 4 ?)
% 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(t1_yellow17,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C,D] :
( m1_subset_1(D,k1_zfmisc_1(k1_funct_1(A,B)))
=> ( ~ r1_xboole_0(k10_relat_1(k3_pralg_3(A,B),k1_tarski(C)),k10_relat_1(k3_pralg_3(A,B),D))
=> r2_hidden(C,D) ) ) ) ).
fof(t2_yellow17,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C,D] :
( ( r2_hidden(D,k1_funct_1(A,C))
& r2_hidden(B,k4_card_3(A)) )
=> r2_hidden(k2_funct_7(B,C,D),k4_card_3(A)) ) ) ) ).
fof(t3_yellow17,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( r2_hidden(B,k1_relat_1(A))
=> ( k4_card_3(A) = k1_xboole_0
| k2_relat_1(k3_pralg_3(A,B)) = k1_funct_1(A,B) ) ) ) ).
fof(t4_yellow17,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( r2_hidden(B,k1_relat_1(A))
=> k10_relat_1(k3_pralg_3(A,B),k1_funct_1(A,B)) = k4_card_3(A) ) ) ).
fof(t5_yellow17,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C,D] :
( ( r2_hidden(D,k1_funct_1(A,C))
& r2_hidden(C,k1_relat_1(A))
& r2_hidden(B,k4_card_3(A)) )
=> r2_hidden(k2_funct_7(B,C,D),k10_relat_1(k3_pralg_3(A,C),k1_tarski(D))) ) ) ) ).
fof(t6_yellow17,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C,D,E,F] :
( m1_subset_1(F,k1_zfmisc_1(k1_funct_1(A,D)))
=> ( ( r2_hidden(E,k1_funct_1(A,C))
& r2_hidden(C,k1_relat_1(A))
& r2_hidden(B,k4_card_3(A)) )
=> ( C = D
| ( r2_hidden(B,k10_relat_1(k3_pralg_3(A,D),F))
<=> r2_hidden(k2_funct_7(B,C,E),k10_relat_1(k3_pralg_3(A,D),F)) ) ) ) ) ) ) ).
fof(t7_yellow17,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C,D,E] :
( m1_subset_1(E,k1_zfmisc_1(k1_funct_1(A,C)))
=> ( ( r2_hidden(D,k1_funct_1(A,B))
& r2_hidden(B,k1_relat_1(A))
& r2_hidden(C,k1_relat_1(A)) )
=> ( k4_card_3(A) = k1_xboole_0
| E = k1_funct_1(A,C)
| ( r1_tarski(k10_relat_1(k3_pralg_3(A,B),k1_tarski(D)),k10_relat_1(k3_pralg_3(A,C),E))
<=> ( B = C
& r2_hidden(D,E) ) ) ) ) ) ) ).
fof(t8_yellow17,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v4_waybel_3(B)
& v1_waybel18(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k3_waybel18(A,B)))
=> k1_funct_1(k6_waybel18(A,B,C),D) = k5_waybel18(A,B,D,C) ) ) ) ) ).
fof(t9_yellow17,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v4_waybel_3(B)
& v1_waybel18(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k4_waybel18(A,B,C)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k4_waybel18(A,B,C))))
=> ( ~ r1_xboole_0(k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,C),k6_waybel18(A,B,C),k1_struct_0(k4_waybel18(A,B,C),D)),k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,C),k6_waybel18(A,B,C),E))
=> r2_hidden(D,E) ) ) ) ) ) ) ).
fof(t10_yellow17,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v4_waybel_3(B)
& v1_waybel18(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_subset_1(C,A)
=> k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,C),k6_waybel18(A,B,C),k2_pre_topc(k4_waybel18(A,B,C))) = k2_pre_topc(k3_waybel18(A,B)) ) ) ) ).
fof(t11_yellow17,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v4_waybel_3(B)
& v1_waybel18(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k4_waybel18(A,B,C)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k3_waybel18(A,B)))
=> r2_hidden(k2_funct_7(E,C,D),k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,C),k6_waybel18(A,B,C),k1_struct_0(k4_waybel18(A,B,C),D))) ) ) ) ) ) ).
fof(t12_yellow17,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v4_waybel_3(B)
& v1_waybel18(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k4_waybel18(A,B,C)))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k4_waybel18(A,B,D))))
=> ( F != k2_pre_topc(k4_waybel18(A,B,D))
=> ( r1_tarski(k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,C),k6_waybel18(A,B,C),k1_struct_0(k4_waybel18(A,B,C),E)),k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,D),k6_waybel18(A,B,D),F))
<=> ( C = D
& r2_hidden(E,F) ) ) ) ) ) ) ) ) ) ).
fof(t13_yellow17,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v4_waybel_3(B)
& v1_waybel18(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k4_waybel18(A,B,C)))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k4_waybel18(A,B,D))))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(k3_waybel18(A,B)))
=> ( C != D
=> ( r2_hidden(G,k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,D),k6_waybel18(A,B,D),F))
<=> r2_hidden(k2_funct_7(G,C,E),k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,D),k6_waybel18(A,B,D),F)) ) ) ) ) ) ) ) ) ) ).
fof(t14_yellow17,axiom,
$true ).
fof(t15_yellow17,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ( v2_compts_1(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ~ ( v1_tops_2(B,A)
& r1_tarski(k2_pre_topc(A),k3_tarski(B))
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ~ ( r1_tarski(C,B)
& r1_tarski(k2_pre_topc(A),k3_tarski(C))
& v1_finset_1(C) ) ) ) ) ) ) ).
fof(t16_yellow17,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m2_cantor_1(B,A)
=> ( v2_compts_1(A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(B))
=> ~ ( r1_tarski(k2_pre_topc(A),k3_tarski(C))
& ! [D] :
( ( v1_finset_1(D)
& m1_subset_1(D,k1_zfmisc_1(C)) )
=> ~ r1_tarski(k2_pre_topc(A),k3_tarski(D)) ) ) ) ) ) ) ).
fof(t17_yellow17,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v4_waybel_3(B)
& v1_waybel18(B)
& m1_pboole(B,A) )
=> ! [C] :
~ ( r2_hidden(C,k2_waybel18(A,B))
& ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k4_waybel18(A,B,D))))
=> ~ ( v3_pre_topc(E,k4_waybel18(A,B,D))
& k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,D),k6_waybel18(A,B,D),E) = C ) ) ) ) ) ) ).
fof(t18_yellow17,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v4_waybel_3(B)
& v1_waybel18(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k4_waybel18(A,B,C)))
=> ! [E] :
~ ( r2_hidden(E,k2_waybel18(A,B))
& r1_tarski(k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,C),k6_waybel18(A,B,C),k1_struct_0(k4_waybel18(A,B,C),D)),E)
& E != k2_pre_topc(k3_waybel18(A,B))
& ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k4_waybel18(A,B,C))))
=> ~ ( F != k2_pre_topc(k4_waybel18(A,B,C))
& r2_hidden(D,F)
& v3_pre_topc(F,k4_waybel18(A,B,C))
& E = k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,C),k6_waybel18(A,B,C),F) ) ) ) ) ) ) ) ).
fof(t20_yellow17,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v4_waybel_3(B)
& v1_waybel18(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k4_waybel18(A,B,C)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k2_waybel18(A,B)))
=> ( ( r1_tarski(k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,C),k6_waybel18(A,B,C),k1_struct_0(k4_waybel18(A,B,C),D)),k3_tarski(E))
& ! [F] :
~ ( r2_hidden(F,k2_waybel18(A,B))
& r2_hidden(F,E)
& r1_tarski(k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,C),k6_waybel18(A,B,C),k1_struct_0(k4_waybel18(A,B,C),D)),F) ) )
=> r1_tarski(k2_pre_topc(k3_waybel18(A,B)),k3_tarski(E)) ) ) ) ) ) ) ).
fof(t21_yellow17,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v4_waybel_3(B)
& v1_waybel18(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k2_waybel18(A,B)))
=> ( ! [E] :
( ( v1_finset_1(E)
& m1_subset_1(E,k1_zfmisc_1(D)) )
=> ~ r1_tarski(k2_pre_topc(k3_waybel18(A,B)),k3_tarski(E)) )
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k4_waybel18(A,B,C)))
=> ! [F] :
( ( v1_finset_1(F)
& m1_subset_1(F,k1_zfmisc_1(D)) )
=> ~ ( r1_tarski(k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,C),k6_waybel18(A,B,C),k1_struct_0(k4_waybel18(A,B,C),E)),k3_tarski(F))
& ! [G] :
~ ( r2_hidden(G,k2_waybel18(A,B))
& r2_hidden(G,F)
& r1_tarski(k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,C),k6_waybel18(A,B,C),k1_struct_0(k4_waybel18(A,B,C),E)),G) ) ) ) ) ) ) ) ) ) ).
fof(t22_yellow17,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v4_waybel_3(B)
& v1_waybel18(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k2_waybel18(A,B)))
=> ( ! [E] :
( ( v1_finset_1(E)
& m1_subset_1(E,k1_zfmisc_1(D)) )
=> ~ r1_tarski(k2_pre_topc(k3_waybel18(A,B)),k3_tarski(E)) )
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k4_waybel18(A,B,C)))
=> ! [F] :
( ( v1_finset_1(F)
& m1_subset_1(F,k1_zfmisc_1(D)) )
=> ~ ( r1_tarski(k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,C),k6_waybel18(A,B,C),k1_struct_0(k4_waybel18(A,B,C),E)),k3_tarski(F))
& ! [G] :
( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(k4_waybel18(A,B,C))))
=> ~ ( G != k2_pre_topc(k4_waybel18(A,B,C))
& r2_hidden(E,G)
& r2_hidden(k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,C),k6_waybel18(A,B,C),G),F)
& v3_pre_topc(G,k4_waybel18(A,B,C)) ) ) ) ) ) ) ) ) ) ) ).
fof(t23_yellow17,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v4_waybel_3(B)
& v1_waybel18(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k2_waybel18(A,B)))
=> ~ ( ! [E] :
( m1_subset_1(E,A)
=> v2_compts_1(k4_waybel18(A,B,E)) )
& ! [E] :
( ( v1_finset_1(E)
& m1_subset_1(E,k1_zfmisc_1(D)) )
=> ~ r1_tarski(k2_pre_topc(k3_waybel18(A,B)),k3_tarski(E)) )
& ! [E] :
( m1_subset_1(E,u1_struct_0(k4_waybel18(A,B,C)))
=> ? [F] :
( v1_finset_1(F)
& m1_subset_1(F,k1_zfmisc_1(D))
& r1_tarski(k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,C),k6_waybel18(A,B,C),k1_struct_0(k4_waybel18(A,B,C),E)),k3_tarski(F)) ) ) ) ) ) ) ) ).
fof(t24_yellow17,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v4_waybel_3(B)
& v1_waybel18(B)
& m1_pboole(B,A) )
=> ( ! [C] :
( m1_subset_1(C,A)
=> v2_compts_1(k4_waybel18(A,B,C)) )
=> v2_compts_1(k3_waybel18(A,B)) ) ) ) ).
fof(s1_yellow17,axiom,
( ! [A] :
( m1_subset_1(A,f1_s1_yellow17)
=> ? [B] :
( m1_subset_1(B,u1_struct_0(k4_waybel18(f1_s1_yellow17,f2_s1_yellow17,A)))
& p1_s1_yellow17(B,A) ) )
=> ? [A] :
( m1_subset_1(A,u1_struct_0(k3_waybel18(f1_s1_yellow17,f2_s1_yellow17)))
& ! [B] :
( m1_subset_1(B,f1_s1_yellow17)
=> p1_s1_yellow17(k5_waybel18(f1_s1_yellow17,f2_s1_yellow17,A,B),B) ) ) ) ).
fof(t19_yellow17,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v4_waybel_3(B)
& v1_waybel18(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_waybel18(A,B,C))))) )
=> ( r1_tarski(k2_pre_topc(k4_waybel18(A,B,C)),k3_tarski(D))
=> r1_tarski(k2_pre_topc(k3_waybel18(A,B)),k3_tarski(a_4_0_yellow17(A,B,C,D))) ) ) ) ) ) ).
fof(fraenkel_a_4_0_yellow17,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(B)
& v4_waybel_3(C)
& v1_waybel18(C)
& m1_pboole(C,B)
& m1_subset_1(D,B)
& ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_waybel18(B,C,D))))) )
=> ( r2_hidden(A,a_4_0_yellow17(B,C,D,E))
<=> ? [F] :
( m2_subset_1(F,k1_zfmisc_1(u1_struct_0(k4_waybel18(B,C,D))),E)
& A = k5_pre_topc(k3_waybel18(B,C),k4_waybel18(B,C,D),k6_waybel18(B,C,D),F) ) ) ) ).
%------------------------------------------------------------------------------