SET007 Axioms: SET007+360.ax
%------------------------------------------------------------------------------
% File : SET007+360 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Finite Topological 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 : fin_topo [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 83 ( 9 unt; 0 def)
% Number of atoms : 414 ( 46 equ)
% Maximal formula atoms : 21 ( 4 avg)
% Number of connectives : 418 ( 87 ~; 5 |; 154 &)
% ( 24 <=>; 148 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 6 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 27 ( 25 usr; 1 prp; 0-3 aty)
% Number of functors : 49 ( 49 usr; 9 con; 0-4 aty)
% Number of variables : 183 ( 168 !; 15 ?)
% 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_fin_topo,axiom,
? [A] :
( l1_fin_topo(A)
& v1_fin_topo(A) ) ).
fof(rc2_fin_topo,axiom,
? [A] :
( l1_fin_topo(A)
& ~ v3_struct_0(A)
& v1_fin_topo(A) ) ).
fof(fc1_fin_topo,axiom,
( ~ v3_struct_0(k3_fin_topo)
& v1_fin_topo(k3_fin_topo) ) ).
fof(rc3_fin_topo,axiom,
? [A] :
( l1_fin_topo(A)
& ~ v3_struct_0(A)
& v6_group_1(A)
& v1_fin_topo(A)
& v2_fin_topo(A) ) ).
fof(t1_fin_topo,axiom,
! [A,B] :
( m2_finseq_1(B,k1_zfmisc_1(A))
=> ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,C)
=> ( r1_xreal_0(k3_finseq_1(B),C)
| r1_tarski(k4_finseq_4(k5_numbers,k1_zfmisc_1(A),B,C),k4_finseq_4(k5_numbers,k1_zfmisc_1(A),B,k1_nat_1(C,np__1))) ) ) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(C,D)
& r1_xreal_0(np__1,C)
& r1_xreal_0(D,k3_finseq_1(B)) )
=> r1_tarski(k4_finseq_4(k5_numbers,k1_zfmisc_1(A),B,C),k4_finseq_4(k5_numbers,k1_zfmisc_1(A),B,D)) ) ) ) ) ) ).
fof(t2_fin_topo,axiom,
! [A,B] :
( m2_finseq_1(B,k1_zfmisc_1(A))
=> ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,C)
=> ( r1_xreal_0(k3_finseq_1(B),C)
| r1_tarski(k4_finseq_4(k5_numbers,k1_zfmisc_1(A),B,C),k4_finseq_4(k5_numbers,k1_zfmisc_1(A),B,k1_nat_1(C,np__1))) ) ) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,C)
& r1_xreal_0(D,k3_finseq_1(B))
& r1_tarski(k4_finseq_4(k5_numbers,k1_zfmisc_1(A),B,D),k4_finseq_4(k5_numbers,k1_zfmisc_1(A),B,C)) )
=> ( r1_xreal_0(D,C)
| ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(C,E)
& r1_xreal_0(E,D) )
=> k4_finseq_4(k5_numbers,k1_zfmisc_1(A),B,D) = k4_finseq_4(k5_numbers,k1_zfmisc_1(A),B,E) ) ) ) ) ) ) ) ) ).
fof(d1_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_fin_topo(A,B) = k8_funct_2(u1_struct_0(A),k1_zfmisc_1(u1_struct_0(A)),u1_fin_topo(A),B) ) ) ).
fof(d2_fin_topo,axiom,
k3_fin_topo = g1_fin_topo(k6_domain_1(k5_numbers,np__0),k2_fin_topo(np__0,k2_subset_1(k6_domain_1(k5_numbers,np__0)))) ).
fof(d3_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ( v2_fin_topo(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> r2_hidden(B,k1_fin_topo(A,B)) ) ) ) ).
fof(t3_fin_topo,axiom,
$true ).
fof(t4_fin_topo,axiom,
$true ).
fof(t5_fin_topo,axiom,
$true ).
fof(t6_fin_topo,axiom,
$true ).
fof(t7_fin_topo,axiom,
v2_fin_topo(k3_fin_topo) ).
fof(t8_fin_topo,axiom,
v6_group_1(k3_fin_topo) ).
fof(d4_fin_topo,axiom,
$true ).
fof(d5_fin_topo,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( r1_fin_topo(A,B)
<=> r1_tarski(u1_struct_0(A),k3_tarski(B)) ) ) ).
fof(t10_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(B,k4_fin_topo(A,C))
<=> ( ~ r1_xboole_0(k1_fin_topo(A,B),C)
& ~ r1_xboole_0(k1_fin_topo(A,B),k3_subset_1(u1_struct_0(A),C)) ) ) ) ) ) ).
fof(d7_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k5_fin_topo(A,B) = k5_subset_1(u1_struct_0(A),B,k4_fin_topo(A,B)) ) ) ).
fof(d8_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k6_fin_topo(A,B) = k5_subset_1(u1_struct_0(A),k3_subset_1(u1_struct_0(A),B),k4_fin_topo(A,B)) ) ) ).
fof(t11_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k4_fin_topo(A,B) = k4_subset_1(u1_struct_0(A),k5_fin_topo(A,B),k6_fin_topo(A,B)) ) ) ).
fof(d12_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k10_fin_topo(A,B) = k6_subset_1(u1_struct_0(A),B,k9_fin_topo(A,B)) ) ) ).
fof(d14_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ( v3_fin_topo(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(C,k1_fin_topo(A,B))
=> r2_hidden(B,k1_fin_topo(A,C)) ) ) ) ) ) ).
fof(t12_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(B,k7_fin_topo(A,C))
<=> r1_tarski(k1_fin_topo(A,B),C) ) ) ) ) ).
fof(t13_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(B,k8_fin_topo(A,C))
<=> ~ r1_xboole_0(k1_fin_topo(A,B),C) ) ) ) ) ).
fof(t14_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(B,k9_fin_topo(A,C))
<=> ( r2_hidden(B,C)
& r1_xboole_0(k6_subset_1(u1_struct_0(A),k1_fin_topo(A,B),k6_domain_1(u1_struct_0(A),B)),C) ) ) ) ) ) ).
fof(t15_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(B,k10_fin_topo(A,C))
<=> ( r2_hidden(B,C)
& ~ r1_xboole_0(k6_subset_1(u1_struct_0(A),k1_fin_topo(A,B),k6_domain_1(u1_struct_0(A),B)),C) ) ) ) ) ) ).
fof(t16_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(B,k11_fin_topo(A,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& r2_hidden(D,C)
& r2_hidden(B,k1_fin_topo(A,D)) ) ) ) ) ) ).
fof(t17_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ( v3_fin_topo(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k8_fin_topo(A,B) = k11_fin_topo(A,B) ) ) ) ).
fof(d15_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v4_fin_topo(B,A)
<=> B = k7_fin_topo(A,B) ) ) ) ).
fof(d16_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v5_fin_topo(B,A)
<=> B = k8_fin_topo(A,B) ) ) ) ).
fof(d17_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v6_fin_topo(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(A)))
=> ~ ( B = k4_subset_1(u1_struct_0(A),C,D)
& C != k1_xboole_0
& D != k1_xboole_0
& r1_xboole_0(C,D)
& r1_xboole_0(k8_fin_topo(A,C),D) ) ) ) ) ) ) ).
fof(t18_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_fin_topo(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> r1_tarski(B,k8_fin_topo(A,B)) ) ) ).
fof(t19_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(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)))
=> ( r1_tarski(B,C)
=> r1_tarski(k8_fin_topo(A,B),k8_fin_topo(A,C)) ) ) ) ) ).
fof(t20_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& v2_fin_topo(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v6_fin_topo(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( r2_hidden(C,B)
& ! [D] :
( m2_finseq_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( ~ r1_xreal_0(k3_finseq_1(D),np__0)
& k4_finseq_4(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),D,np__1) = k6_domain_1(u1_struct_0(A),C)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(E,np__0)
& ~ r1_xreal_0(k3_finseq_1(D),E)
& k4_finseq_4(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),D,k1_nat_1(E,np__1)) != k5_subset_1(u1_struct_0(A),k8_fin_topo(A,k4_finseq_4(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),D,E)),B) ) )
& r1_tarski(B,k4_finseq_4(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),D,k3_finseq_1(D))) ) ) ) ) ) ) ) ).
fof(t21_fin_topo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,A)
=> ( r2_hidden(C,k3_subset_1(A,B))
<=> ~ r2_hidden(C,B) ) ) ) ) ).
fof(t22_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k3_subset_1(u1_struct_0(A),k7_fin_topo(A,k3_subset_1(u1_struct_0(A),B))) = k8_fin_topo(A,B) ) ) ).
fof(t23_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k3_subset_1(u1_struct_0(A),k8_fin_topo(A,k3_subset_1(u1_struct_0(A),B))) = k7_fin_topo(A,B) ) ) ).
fof(t24_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k4_fin_topo(A,B) = k5_subset_1(u1_struct_0(A),k8_fin_topo(A,B),k8_fin_topo(A,k3_subset_1(u1_struct_0(A),B))) ) ) ).
fof(t25_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k4_fin_topo(A,k3_subset_1(u1_struct_0(A),B)) = k4_fin_topo(A,B) ) ) ).
fof(t26_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(B,k9_fin_topo(A,C))
& r2_hidden(B,k8_fin_topo(A,k6_subset_1(u1_struct_0(A),C,k6_domain_1(u1_struct_0(A),B)))) ) ) ) ) ).
fof(t27_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( k9_fin_topo(A,B) != k1_xboole_0
& k1_card_1(B) != np__1
& v6_fin_topo(B,A) ) ) ) ).
fof(t28_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_fin_topo(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> r1_tarski(k7_fin_topo(A,B),B) ) ) ).
fof(t29_fin_topo,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( B = C
<=> k3_subset_1(A,B) = k3_subset_1(A,C) ) ) ) ).
fof(t30_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v4_fin_topo(B,A)
=> v5_fin_topo(k3_subset_1(u1_struct_0(A),B),A) ) ) ) ).
fof(t31_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v5_fin_topo(B,A)
=> v4_fin_topo(k3_subset_1(u1_struct_0(A),B),A) ) ) ) ).
fof(s1_fin_topo,axiom,
( ( v1_finset_1(f2_s1_fin_topo)
& r1_tarski(f3_s1_fin_topo,f2_s1_fin_topo)
& ! [A] :
( m1_subset_1(A,k1_zfmisc_1(f1_s1_fin_topo))
=> ( r1_tarski(A,f2_s1_fin_topo)
=> ( r1_tarski(A,f4_s1_fin_topo(A))
& r1_tarski(f4_s1_fin_topo(A),f2_s1_fin_topo) ) ) ) )
=> ? [A] :
( m2_finseq_1(A,k1_zfmisc_1(f1_s1_fin_topo))
& ~ r1_xreal_0(k3_finseq_1(A),np__0)
& k4_finseq_4(k5_numbers,k1_zfmisc_1(f1_s1_fin_topo),A,np__1) = f3_s1_fin_topo
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(B,np__0)
& ~ r1_xreal_0(k3_finseq_1(A),B)
& k4_finseq_4(k5_numbers,k1_zfmisc_1(f1_s1_fin_topo),A,k1_nat_1(B,np__1)) != f4_s1_fin_topo(k4_finseq_4(k5_numbers,k1_zfmisc_1(f1_s1_fin_topo),A,B)) ) )
& f4_s1_fin_topo(k4_finseq_4(k5_numbers,k1_zfmisc_1(f1_s1_fin_topo),A,k3_finseq_1(A))) = k4_finseq_4(k5_numbers,k1_zfmisc_1(f1_s1_fin_topo),A,k3_finseq_1(A))
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(C,k3_finseq_1(A))
=> ( r1_xreal_0(B,np__0)
| r1_xreal_0(C,B)
| ( r1_tarski(k4_finseq_4(k5_numbers,k1_zfmisc_1(f1_s1_fin_topo),A,B),f2_s1_fin_topo)
& r2_xboole_0(k4_finseq_4(k5_numbers,k1_zfmisc_1(f1_s1_fin_topo),A,B),k4_finseq_4(k5_numbers,k1_zfmisc_1(f1_s1_fin_topo),A,C)) ) ) ) ) ) ) ) ).
fof(dt_l1_fin_topo,axiom,
! [A] :
( l1_fin_topo(A)
=> l1_struct_0(A) ) ).
fof(existence_l1_fin_topo,axiom,
? [A] : l1_fin_topo(A) ).
fof(abstractness_v1_fin_topo,axiom,
! [A] :
( l1_fin_topo(A)
=> ( v1_fin_topo(A)
=> A = g1_fin_topo(u1_struct_0(A),u1_fin_topo(A)) ) ) ).
fof(dt_k1_fin_topo,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_subset_1(k1_fin_topo(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k2_fin_topo,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_tarski(A)))
=> ( v1_funct_1(k2_fin_topo(A,B))
& v1_funct_2(k2_fin_topo(A,B),k1_tarski(A),k1_zfmisc_1(k1_tarski(A)))
& m2_relset_1(k2_fin_topo(A,B),k1_tarski(A),k1_zfmisc_1(k1_tarski(A))) ) ) ).
fof(redefinition_k2_fin_topo,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_tarski(A)))
=> k2_fin_topo(A,B) = k3_cqc_lang(A,B) ) ).
fof(dt_k3_fin_topo,axiom,
( v1_fin_topo(k3_fin_topo)
& l1_fin_topo(k3_fin_topo) ) ).
fof(dt_k4_fin_topo,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k4_fin_topo(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k5_fin_topo,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k5_fin_topo(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k6_fin_topo,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k6_fin_topo(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k7_fin_topo,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k7_fin_topo(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k8_fin_topo,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k8_fin_topo(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k9_fin_topo,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k9_fin_topo(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k10_fin_topo,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k10_fin_topo(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k11_fin_topo,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k11_fin_topo(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k12_fin_topo,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k12_fin_topo(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k13_fin_topo,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k13_fin_topo(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_u1_fin_topo,axiom,
! [A] :
( l1_fin_topo(A)
=> ( v1_funct_1(u1_fin_topo(A))
& v1_funct_2(u1_fin_topo(A),u1_struct_0(A),k1_zfmisc_1(u1_struct_0(A)))
& m2_relset_1(u1_fin_topo(A),u1_struct_0(A),k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(dt_g1_fin_topo,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_zfmisc_1(A))
& m1_relset_1(B,A,k1_zfmisc_1(A)) )
=> ( v1_fin_topo(g1_fin_topo(A,B))
& l1_fin_topo(g1_fin_topo(A,B)) ) ) ).
fof(free_g1_fin_topo,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_zfmisc_1(A))
& m1_relset_1(B,A,k1_zfmisc_1(A)) )
=> ! [C,D] :
( g1_fin_topo(A,B) = g1_fin_topo(C,D)
=> ( A = C
& B = D ) ) ) ).
fof(t9_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_fin_topo(A)
& l1_fin_topo(A) )
=> r1_fin_topo(A,a_1_0_fin_topo(A)) ) ).
fof(d6_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k4_fin_topo(A,B) = a_2_0_fin_topo(A,B) ) ) ).
fof(d9_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k7_fin_topo(A,B) = a_2_1_fin_topo(A,B) ) ) ).
fof(d10_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k8_fin_topo(A,B) = a_2_2_fin_topo(A,B) ) ) ).
fof(d11_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k9_fin_topo(A,B) = a_2_3_fin_topo(A,B) ) ) ).
fof(d13_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k11_fin_topo(A,B) = a_2_4_fin_topo(A,B) ) ) ).
fof(d18_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k12_fin_topo(A,B) = k1_setfam_1(a_2_5_fin_topo(A,B)) ) ) ).
fof(d19_fin_topo,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k13_fin_topo(A,B) = k3_tarski(a_2_6_fin_topo(A,B)) ) ) ).
fof(fraenkel_a_1_0_fin_topo,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v2_fin_topo(B)
& l1_fin_topo(B) )
=> ( r2_hidden(A,a_1_0_fin_topo(B))
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(B))
& A = k1_fin_topo(B,C) ) ) ) ).
fof(fraenkel_a_2_0_fin_topo,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& l1_fin_topo(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_2_0_fin_topo(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& A = D
& ~ r1_xboole_0(k1_fin_topo(B,D),C)
& ~ r1_xboole_0(k1_fin_topo(B,D),k3_subset_1(u1_struct_0(B),C)) ) ) ) ).
fof(fraenkel_a_2_1_fin_topo,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& l1_fin_topo(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_2_1_fin_topo(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& A = D
& r1_tarski(k1_fin_topo(B,D),C) ) ) ) ).
fof(fraenkel_a_2_2_fin_topo,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& l1_fin_topo(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_2_2_fin_topo(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& A = D
& ~ r1_xboole_0(k1_fin_topo(B,D),C) ) ) ) ).
fof(fraenkel_a_2_3_fin_topo,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& l1_fin_topo(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_2_3_fin_topo(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& A = D
& r2_hidden(D,C)
& r1_xboole_0(k6_subset_1(u1_struct_0(B),k1_fin_topo(B,D),k6_domain_1(u1_struct_0(B),D)),C) ) ) ) ).
fof(fraenkel_a_2_4_fin_topo,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& l1_fin_topo(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_2_4_fin_topo(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& A = D
& ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& r2_hidden(E,C)
& r2_hidden(D,k1_fin_topo(B,E)) ) ) ) ) ).
fof(fraenkel_a_2_5_fin_topo,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& l1_fin_topo(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_2_5_fin_topo(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
& A = D
& r1_tarski(C,D)
& v5_fin_topo(D,B) ) ) ) ).
fof(fraenkel_a_2_6_fin_topo,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& l1_fin_topo(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_2_6_fin_topo(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
& A = D
& r1_tarski(C,D)
& v4_fin_topo(D,B) ) ) ) ).
%------------------------------------------------------------------------------