SET007 Axioms: SET007+217.ax
%------------------------------------------------------------------------------
% File : SET007+217 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Compact Spaces
% 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 : compts_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 33 ( 8 unt; 0 def)
% Number of atoms : 232 ( 21 equ)
% Maximal formula atoms : 16 ( 7 avg)
% Number of connectives : 229 ( 30 ~; 2 |; 94 &)
% ( 13 <=>; 90 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 9 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 30 ( 28 usr; 1 prp; 0-3 aty)
% Number of functors : 15 ( 15 usr; 1 con; 0-4 aty)
% Number of variables : 74 ( 74 !; 0 ?)
% 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(d1_compts_1,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r1_compts_1(A,B,C)
<=> r1_tarski(C,k5_setfam_1(u1_struct_0(A),B)) ) ) ) ) ).
fof(d2_compts_1,axiom,
! [A] :
( v1_compts_1(A)
<=> ( A != k1_xboole_0
& ! [B] :
~ ( B != k1_xboole_0
& r1_tarski(B,A)
& v1_finset_1(B)
& k1_setfam_1(B) = k1_xboole_0 ) ) ) ).
fof(d3_compts_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( v2_compts_1(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ~ ( r1_pre_topc(A,B)
& v1_tops_2(B,A)
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ~ ( r1_tarski(C,B)
& r1_pre_topc(A,C)
& v1_finset_1(C) ) ) ) ) ) ) ).
fof(d4_compts_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( v3_compts_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( B != C
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v3_pre_topc(D,A)
& v3_pre_topc(E,A)
& r2_hidden(B,D)
& r2_hidden(C,E)
& r1_xboole_0(D,E) ) ) ) ) ) ) ) ) ).
fof(d5_compts_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( v4_compts_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( C != k1_xboole_0
& v4_pre_topc(C,A)
& r2_hidden(B,k3_subset_1(u1_struct_0(A),C))
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v3_pre_topc(D,A)
& v3_pre_topc(E,A)
& r2_hidden(B,D)
& r1_tarski(C,E)
& r1_xboole_0(D,E) ) ) ) ) ) ) ) ) ).
fof(d6_compts_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( v5_compts_1(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)))
=> ~ ( B != k1_xboole_0
& C != k1_xboole_0
& v4_pre_topc(B,A)
& v4_pre_topc(C,A)
& r1_xboole_0(B,C)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v3_pre_topc(D,A)
& v3_pre_topc(E,A)
& r1_tarski(B,D)
& r1_tarski(C,E)
& r1_xboole_0(D,E) ) ) ) ) ) ) ) ) ).
fof(d7_compts_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v6_compts_1(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ~ ( r1_compts_1(A,C,B)
& v1_tops_2(C,A)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ~ ( r1_tarski(D,C)
& r1_compts_1(A,D,B)
& v1_finset_1(D) ) ) ) ) ) ) ) ).
fof(t1_compts_1,axiom,
$true ).
fof(t2_compts_1,axiom,
$true ).
fof(t3_compts_1,axiom,
$true ).
fof(t4_compts_1,axiom,
$true ).
fof(t5_compts_1,axiom,
$true ).
fof(t6_compts_1,axiom,
$true ).
fof(t7_compts_1,axiom,
$true ).
fof(t8_compts_1,axiom,
$true ).
fof(t9_compts_1,axiom,
! [A] :
( l1_pre_topc(A)
=> v6_compts_1(k1_pre_topc(A),A) ) ).
fof(t10_compts_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( v2_compts_1(A)
<=> v6_compts_1(k2_pre_topc(A),A) ) ) ).
fof(t11_compts_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_pre_topc(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r1_tarski(C,k2_pre_topc(B))
=> ( v6_compts_1(C,A)
<=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ( D = C
=> v6_compts_1(D,B) ) ) ) ) ) ) ) ).
fof(t12_compts_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( B = k1_xboole_0
=> ( v6_compts_1(B,A)
<=> v2_compts_1(k3_pre_topc(A,B)) ) )
& ( v2_pre_topc(A)
=> ( B = k1_xboole_0
| ( v6_compts_1(B,A)
<=> v2_compts_1(k3_pre_topc(A,B)) ) ) ) ) ) ) ).
fof(t13_compts_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(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_compts_1(B)
& v2_tops_2(B,A)
& k6_setfam_1(u1_struct_0(A),B) = k1_xboole_0 ) ) ) ) ).
fof(t14_compts_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v2_compts_1(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ~ ( B != k1_xboole_0
& v2_tops_2(B,A)
& k6_setfam_1(u1_struct_0(A),B) = k1_xboole_0
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ~ ( C != k1_xboole_0
& r1_tarski(C,B)
& v1_finset_1(C)
& k6_setfam_1(u1_struct_0(A),C) = k1_xboole_0 ) ) ) ) ) ) ).
fof(t15_compts_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v3_compts_1(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v6_compts_1(B,A)
=> ( B = k1_xboole_0
| ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( r2_hidden(C,k3_subset_1(u1_struct_0(A),B))
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v3_pre_topc(D,A)
& v3_pre_topc(E,A)
& r2_hidden(C,D)
& r1_tarski(B,E)
& r1_xboole_0(D,E) ) ) ) ) ) ) ) ) ) ) ).
fof(t16_compts_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v3_compts_1(A)
& v6_compts_1(B,A) )
=> v4_pre_topc(B,A) ) ) ) ).
fof(t17_compts_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v2_compts_1(A)
& v4_pre_topc(B,A) )
=> v6_compts_1(B,A) ) ) ) ).
fof(t18_compts_1,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)))
=> ( ( v6_compts_1(B,A)
& r1_tarski(C,B)
& v4_pre_topc(C,A) )
=> v6_compts_1(C,A) ) ) ) ) ).
fof(t19_compts_1,axiom,
! [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)))
=> ( ( v6_compts_1(B,A)
& v6_compts_1(C,A) )
=> v6_compts_1(k4_subset_1(u1_struct_0(A),B,C),A) ) ) ) ) ).
fof(t20_compts_1,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)))
=> ( ( v3_compts_1(A)
& v6_compts_1(B,A)
& v6_compts_1(C,A) )
=> v6_compts_1(k5_subset_1(u1_struct_0(A),B,C),A) ) ) ) ) ).
fof(t21_compts_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ( v3_compts_1(A)
& v2_compts_1(A) )
=> v4_compts_1(A) ) ) ).
fof(t22_compts_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ( v3_compts_1(A)
& v2_compts_1(A) )
=> v5_compts_1(A) ) ) ).
fof(t23_compts_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( ( v2_compts_1(A)
& v5_pre_topc(C,A,B)
& k2_relat_1(C) = k2_pre_topc(B) )
=> v2_compts_1(B) ) ) ) ) ).
fof(t24_compts_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( ( ~ v3_struct_0(C)
& l1_pre_topc(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(C))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(C)) )
=> ( ( v5_pre_topc(D,A,C)
& k2_relat_1(D) = k2_pre_topc(C)
& v6_compts_1(B,A) )
=> v6_compts_1(k4_pre_topc(A,C,D,B),C) ) ) ) ) ) ).
fof(t25_compts_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( ( v2_compts_1(A)
& v3_compts_1(B)
& k2_relat_1(C) = k2_pre_topc(B)
& v5_pre_topc(C,A,B) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( v4_pre_topc(D,A)
=> v4_pre_topc(k4_pre_topc(A,B,C,D),B) ) ) ) ) ) ) ).
fof(t26_compts_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( ( v2_compts_1(A)
& v3_compts_1(B)
& k1_relat_1(C) = k2_pre_topc(A)
& k2_relat_1(C) = k2_pre_topc(B)
& v2_funct_1(C)
& v5_pre_topc(C,A,B) )
=> v3_tops_2(C,A,B) ) ) ) ) ).
%------------------------------------------------------------------------------