SET007 Axioms: SET007+673.ax
%------------------------------------------------------------------------------
% File : SET007+673 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Formal 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 : fintopo2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 92 ( 2 unt; 0 def)
% Number of atoms : 571 ( 73 equ)
% Maximal formula atoms : 14 ( 6 avg)
% Number of connectives : 588 ( 109 ~; 0 |; 200 &)
% ( 32 <=>; 247 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 21 ( 19 usr; 1 prp; 0-3 aty)
% Number of functors : 42 ( 42 usr; 3 con; 0-4 aty)
% Number of variables : 276 ( 255 !; 21 ?)
% 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_fintopo2,axiom,
? [A] :
( l1_fintopo2(A)
& v1_fintopo2(A) ) ).
fof(rc2_fintopo2,axiom,
? [A] :
( l1_fintopo2(A)
& ~ v3_struct_0(A)
& v1_fintopo2(A) ) ).
fof(t1_fintopo2,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(k7_fin_topo(A,B),k7_fin_topo(A,C)) ) ) ) ) ).
fof(t2_fintopo2,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),k3_subset_1(u1_struct_0(A),k7_fin_topo(A,B))) ) ) ).
fof(t3_fintopo2,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) = k6_subset_1(u1_struct_0(A),k8_fin_topo(A,B),k7_fin_topo(A,B)) ) ) ).
fof(t4_fintopo2,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(k3_subset_1(u1_struct_0(A),B),A)
=> v5_fin_topo(B,A) ) ) ) ).
fof(t5_fintopo2,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(k3_subset_1(u1_struct_0(A),B),A)
=> v4_fin_topo(B,A) ) ) ) ).
fof(d1_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( ( r2_hidden(C,k1_fin_topo(A,B))
& r2_hidden(C,D) )
=> k1_fintopo2(A,B,C,D) = k8_margrel1 )
& ( ~ ( r2_hidden(C,k1_fin_topo(A,B))
& r2_hidden(C,D) )
=> k1_fintopo2(A,B,C,D) = k7_margrel1 ) ) ) ) ) ) ).
fof(d2_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( ( r2_hidden(C,k1_fin_topo(A,B))
& r2_hidden(C,k3_subset_1(u1_struct_0(A),D)) )
=> k2_fintopo2(A,B,C,D) = k8_margrel1 )
& ( ~ ( r2_hidden(C,k1_fin_topo(A,B))
& r2_hidden(C,k3_subset_1(u1_struct_0(A),D)) )
=> k2_fintopo2(A,B,C,D) = k7_margrel1 ) ) ) ) ) ) ).
fof(t6_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( k1_fintopo2(A,B,C,D) = k8_margrel1
<=> ( r2_hidden(C,k1_fin_topo(A,B))
& r2_hidden(C,D) ) ) ) ) ) ) ).
fof(t7_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( k2_fintopo2(A,B,C,D) = k8_margrel1
<=> ( r2_hidden(C,k1_fin_topo(A,B))
& r2_hidden(C,k3_subset_1(u1_struct_0(A),D)) ) ) ) ) ) ) ).
fof(t8_fintopo2,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))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& ? [E] :
( m1_subset_1(E,u1_struct_0(A))
& k1_fintopo2(A,B,D,C) = k8_margrel1
& k2_fintopo2(A,B,E,C) = k8_margrel1 ) ) ) ) ) ) ).
fof(d3_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_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))
=> k3_fintopo2(A,B,C) = k8_margrel1 )
& ( ~ r2_hidden(C,k1_fin_topo(A,B))
=> k3_fintopo2(A,B,C) = k7_margrel1 ) ) ) ) ) ).
fof(t9_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k3_fintopo2(A,B,C) = k8_margrel1
<=> r2_hidden(C,k1_fin_topo(A,B)) ) ) ) ) ).
fof(t10_fintopo2,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))
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( k3_fintopo2(A,B,D) = k8_margrel1
=> k1_fintopo2(A,B,D,C) = k8_margrel1 ) ) ) ) ) ) ).
fof(t11_fintopo2,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))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& k1_fintopo2(A,B,D,C) = k8_margrel1 ) ) ) ) ) ).
fof(d4_fintopo2,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,C)
=> k4_fintopo2(A,B,C) = k8_margrel1 )
& ( ~ r2_hidden(B,C)
=> k4_fintopo2(A,B,C) = k7_margrel1 ) ) ) ) ) ).
fof(t12_fintopo2,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)))
=> ( k4_fintopo2(A,B,C) = k8_margrel1
<=> r2_hidden(B,C) ) ) ) ) ).
fof(t13_fintopo2,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,k5_fin_topo(A,C))
<=> ( ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& ? [E] :
( m1_subset_1(E,u1_struct_0(A))
& k1_fintopo2(A,B,D,C) = k8_margrel1
& k2_fintopo2(A,B,E,C) = k8_margrel1 ) )
& k4_fintopo2(A,B,C) = k8_margrel1 ) ) ) ) ) ).
fof(t14_fintopo2,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,k6_fin_topo(A,C))
<=> ( ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& ? [E] :
( m1_subset_1(E,u1_struct_0(A))
& k1_fintopo2(A,B,D,C) = k8_margrel1
& k2_fintopo2(A,B,E,C) = k8_margrel1 ) )
& k4_fintopo2(A,B,C) = k7_margrel1 ) ) ) ) ) ).
fof(d5_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ( B = C
=> k5_fintopo2(A,B,C) = k8_margrel1 )
& ( B != C
=> k5_fintopo2(A,B,C) = k7_margrel1 ) ) ) ) ) ).
fof(t15_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k5_fintopo2(A,B,C) = k8_margrel1
<=> B = C ) ) ) ) ).
fof(t16_fintopo2,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))
<=> ( k4_fintopo2(A,B,C) = k8_margrel1
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( k1_fintopo2(A,B,D,C) = k8_margrel1
& k5_fintopo2(A,B,D) = k7_margrel1 ) ) ) ) ) ) ) ).
fof(t17_fintopo2,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))
<=> ( k4_fintopo2(A,B,C) = k8_margrel1
& ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& k1_fintopo2(A,B,D,C) = k8_margrel1
& k5_fintopo2(A,B,D) = k7_margrel1 ) ) ) ) ) ) ).
fof(t18_fintopo2,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))
& k4_fintopo2(A,D,C) = k8_margrel1
& k3_fintopo2(A,D,B) = k8_margrel1 ) ) ) ) ) ).
fof(d6_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k6_fintopo2(A,B) = k8_funct_2(u1_struct_0(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))),u1_fintopo2(A),B) ) ) ).
fof(d8_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A) )
=> ( v2_fintopo2(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(C,k6_fintopo2(A,B))
=> r2_hidden(B,C) ) ) ) ) ) ).
fof(t19_fintopo2,axiom,
$true ).
fof(t20_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(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_fintopo2(A,C))
<=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(D,k6_fintopo2(A,B))
=> ( ~ r1_xboole_0(D,C)
& ~ r1_xboole_0(D,k3_subset_1(u1_struct_0(A),C)) ) ) ) ) ) ) ) ).
fof(t21_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(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_fintopo2(A,C))
<=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(D,k6_fintopo2(A,B))
& r1_xboole_0(D,C) ) ) ) ) ) ) ).
fof(t22_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(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_fintopo2(A,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
& r2_hidden(D,k6_fintopo2(A,B))
& r1_tarski(D,C) ) ) ) ) ) ).
fof(t23_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(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_fintopo2(A,C))
<=> ( r2_hidden(B,C)
& ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
& r2_hidden(D,k6_fintopo2(A,B))
& r1_xboole_0(k6_subset_1(u1_struct_0(A),D,k1_struct_0(A,B)),C) ) ) ) ) ) ) ).
fof(d13_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k12_fintopo2(A,B) = k6_subset_1(u1_struct_0(A),B,k11_fintopo2(A,B)) ) ) ).
fof(t24_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(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,k12_fintopo2(A,C))
<=> ( r2_hidden(B,C)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(D,k6_fintopo2(A,B))
& r1_xboole_0(k6_subset_1(u1_struct_0(A),D,k1_struct_0(A,B)),C) ) ) ) ) ) ) ) ).
fof(t25_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(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(k9_fintopo2(A,B),k9_fintopo2(A,C)) ) ) ) ) ).
fof(t26_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(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(k10_fintopo2(A,B),k10_fintopo2(A,C)) ) ) ) ) ).
fof(t27_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(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(k4_subset_1(u1_struct_0(A),k9_fintopo2(A,B),k9_fintopo2(A,C)),k9_fintopo2(A,k4_subset_1(u1_struct_0(A),B,C))) ) ) ) ).
fof(t28_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(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(k9_fintopo2(A,k5_subset_1(u1_struct_0(A),B,C)),k5_subset_1(u1_struct_0(A),k9_fintopo2(A,B),k9_fintopo2(A,C))) ) ) ) ).
fof(t29_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(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(k4_subset_1(u1_struct_0(A),k10_fintopo2(A,B),k10_fintopo2(A,C)),k10_fintopo2(A,k4_subset_1(u1_struct_0(A),B,C))) ) ) ) ).
fof(t30_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(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(k10_fintopo2(A,k5_subset_1(u1_struct_0(A),B,C)),k5_subset_1(u1_struct_0(A),k10_fintopo2(A,B),k10_fintopo2(A,C))) ) ) ) ).
fof(t31_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A) )
=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(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)))
=> ~ ( r2_hidden(C,k6_fintopo2(A,B))
& r2_hidden(D,k6_fintopo2(A,B))
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(E,k6_fintopo2(A,B))
& r1_tarski(E,k5_subset_1(u1_struct_0(A),C,D)) ) ) ) ) ) )
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> k9_fintopo2(A,k4_subset_1(u1_struct_0(A),B,C)) = k4_subset_1(u1_struct_0(A),k9_fintopo2(A,B),k9_fintopo2(A,C)) ) ) ) ) ).
fof(t32_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A) )
=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(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)))
=> ~ ( r2_hidden(C,k6_fintopo2(A,B))
& r2_hidden(D,k6_fintopo2(A,B))
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(E,k6_fintopo2(A,B))
& r1_tarski(E,k5_subset_1(u1_struct_0(A),C,D)) ) ) ) ) ) )
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> k5_subset_1(u1_struct_0(A),k10_fintopo2(A,B),k10_fintopo2(A,C)) = k10_fintopo2(A,k5_subset_1(u1_struct_0(A),B,C)) ) ) ) ) ).
fof(t33_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(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)))
=> ( ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(E,k6_fintopo2(A,D))
& r2_hidden(F,k6_fintopo2(A,D))
& ! [G] :
( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(G,k6_fintopo2(A,D))
& r1_tarski(G,k5_subset_1(u1_struct_0(A),E,F)) ) ) ) ) ) )
=> r1_tarski(k8_fintopo2(A,k4_subset_1(u1_struct_0(A),B,C)),k4_subset_1(u1_struct_0(A),k8_fintopo2(A,B),k8_fintopo2(A,C))) ) ) ) ) ).
fof(t34_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A) )
=> ( ( v2_fintopo2(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)))
=> k8_fintopo2(A,k4_subset_1(u1_struct_0(A),B,C)) = k4_subset_1(u1_struct_0(A),k8_fintopo2(A,B),k8_fintopo2(A,C)) ) ) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(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)))
=> ~ ( r2_hidden(C,k6_fintopo2(A,B))
& r2_hidden(D,k6_fintopo2(A,B))
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(E,k6_fintopo2(A,B))
& r1_tarski(E,k5_subset_1(u1_struct_0(A),C,D)) ) ) ) ) ) ) ) ) ).
fof(t35_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(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_fintopo2(A,C))
<=> ( r2_hidden(B,C)
& ~ r2_hidden(B,k9_fintopo2(A,k6_subset_1(u1_struct_0(A),C,k1_struct_0(A,B)))) ) ) ) ) ) ).
fof(t36_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A) )
=> ( v2_fintopo2(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> r1_tarski(B,k9_fintopo2(A,B)) ) ) ) ).
fof(t37_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A) )
=> ( v2_fintopo2(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> r1_tarski(k10_fintopo2(A,B),B) ) ) ) ).
fof(t38_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k3_subset_1(u1_struct_0(A),k9_fintopo2(A,k3_subset_1(u1_struct_0(A),B))) = k10_fintopo2(A,B) ) ) ).
fof(t39_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k3_subset_1(u1_struct_0(A),k10_fintopo2(A,k3_subset_1(u1_struct_0(A),B))) = k9_fintopo2(A,B) ) ) ).
fof(t40_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k8_fintopo2(A,B) = k5_subset_1(u1_struct_0(A),k9_fintopo2(A,B),k9_fintopo2(A,k3_subset_1(u1_struct_0(A),B))) ) ) ).
fof(t41_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k8_fintopo2(A,B) = k5_subset_1(u1_struct_0(A),k9_fintopo2(A,B),k3_subset_1(u1_struct_0(A),k10_fintopo2(A,B))) ) ) ).
fof(t42_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k8_fintopo2(A,B) = k8_fintopo2(A,k3_subset_1(u1_struct_0(A),B)) ) ) ).
fof(t43_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k8_fintopo2(A,B) = k6_subset_1(u1_struct_0(A),k9_fintopo2(A,B),k10_fintopo2(A,B)) ) ) ).
fof(d14_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k13_fintopo2(A,B) = k5_subset_1(u1_struct_0(A),B,k8_fintopo2(A,B)) ) ) ).
fof(d15_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k14_fintopo2(A,B) = k5_subset_1(u1_struct_0(A),k3_subset_1(u1_struct_0(A),B),k8_fintopo2(A,B)) ) ) ).
fof(t44_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k8_fintopo2(A,B) = k4_subset_1(u1_struct_0(A),k13_fintopo2(A,B),k14_fintopo2(A,B)) ) ) ).
fof(d16_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v3_fintopo2(B,A)
<=> B = k10_fintopo2(A,B) ) ) ) ).
fof(d17_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v4_fintopo2(B,A)
<=> B = k9_fintopo2(A,B) ) ) ) ).
fof(t45_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v3_fintopo2(k3_subset_1(u1_struct_0(A),B),A)
=> v4_fintopo2(B,A) ) ) ) ).
fof(t46_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v4_fintopo2(k3_subset_1(u1_struct_0(A),B),A)
=> v3_fintopo2(B,A) ) ) ) ).
fof(t47_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(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)))
=> ( ( v2_fintopo2(A)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> r2_hidden(k1_struct_0(A,D),k6_fintopo2(A,D)) ) )
=> k9_fintopo2(A,k5_subset_1(u1_struct_0(A),B,C)) = k5_subset_1(u1_struct_0(A),k9_fintopo2(A,B),k9_fintopo2(A,C)) ) ) ) ) ).
fof(t48_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(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)))
=> ( ( v2_fintopo2(A)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> r2_hidden(k1_struct_0(A,D),k6_fintopo2(A,D)) ) )
=> k4_subset_1(u1_struct_0(A),k10_fintopo2(A,B),k10_fintopo2(A,C)) = k10_fintopo2(A,k4_subset_1(u1_struct_0(A),B,C)) ) ) ) ) ).
fof(dt_l1_fintopo2,axiom,
! [A] :
( l1_fintopo2(A)
=> l1_struct_0(A) ) ).
fof(existence_l1_fintopo2,axiom,
? [A] : l1_fintopo2(A) ).
fof(abstractness_v1_fintopo2,axiom,
! [A] :
( l1_fintopo2(A)
=> ( v1_fintopo2(A)
=> A = g1_fintopo2(u1_struct_0(A),u1_fintopo2(A)) ) ) ).
fof(dt_k1_fintopo2,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k1_fintopo2(A,B,C,D),k6_margrel1) ) ).
fof(dt_k2_fintopo2,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k2_fintopo2(A,B,C,D),k6_margrel1) ) ).
fof(dt_k3_fintopo2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k3_fintopo2(A,B,C),k6_margrel1) ) ).
fof(dt_k4_fintopo2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k4_fintopo2(A,B,C),k6_margrel1) ) ).
fof(dt_k5_fintopo2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k5_fintopo2(A,B,C),k6_margrel1) ) ).
fof(dt_k6_fintopo2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_subset_1(k6_fintopo2(A,B),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ).
fof(dt_k7_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ( ~ v3_struct_0(k7_fintopo2(A))
& v1_fintopo2(k7_fintopo2(A))
& l1_fintopo2(k7_fintopo2(A)) ) ) ).
fof(dt_k8_fintopo2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k8_fintopo2(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k9_fintopo2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k9_fintopo2(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k10_fintopo2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k10_fintopo2(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k11_fintopo2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k11_fintopo2(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k12_fintopo2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k12_fintopo2(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k13_fintopo2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k13_fintopo2(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k14_fintopo2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k14_fintopo2(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_u1_fintopo2,axiom,
! [A] :
( l1_fintopo2(A)
=> ( v1_funct_1(u1_fintopo2(A))
& v1_funct_2(u1_fintopo2(A),u1_struct_0(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
& m2_relset_1(u1_fintopo2(A),u1_struct_0(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ) ).
fof(dt_g1_fintopo2,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_zfmisc_1(k1_zfmisc_1(A)))
& m1_relset_1(B,A,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( v1_fintopo2(g1_fintopo2(A,B))
& l1_fintopo2(g1_fintopo2(A,B)) ) ) ).
fof(free_g1_fintopo2,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_zfmisc_1(k1_zfmisc_1(A)))
& m1_relset_1(B,A,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C,D] :
( g1_fintopo2(A,B) = g1_fintopo2(C,D)
=> ( A = C
& B = D ) ) ) ).
fof(d7_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_fintopo2(B)
& l1_fintopo2(B) )
=> ( B = k7_fintopo2(A)
<=> ( u1_struct_0(B) = u1_struct_0(A)
& ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> k6_fintopo2(B,C) = a_3_0_fintopo2(A,B,C) ) ) ) ) ) ).
fof(d9_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k8_fintopo2(A,B) = a_2_0_fintopo2(A,B) ) ) ).
fof(d10_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k9_fintopo2(A,B) = a_2_1_fintopo2(A,B) ) ) ).
fof(d11_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k10_fintopo2(A,B) = a_2_2_fintopo2(A,B) ) ) ).
fof(d12_fintopo2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fintopo2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k11_fintopo2(A,B) = a_2_3_fintopo2(A,B) ) ) ).
fof(fraenkel_a_3_0_fintopo2,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& l1_pre_topc(B)
& ~ v3_struct_0(C)
& v1_fintopo2(C)
& l1_fintopo2(C)
& m1_subset_1(D,u1_struct_0(C)) )
=> ( r2_hidden(A,a_3_0_fintopo2(B,C,D))
<=> ? [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
& A = E
& r2_hidden(E,u1_pre_topc(B))
& r2_hidden(D,E) ) ) ) ).
fof(fraenkel_a_2_0_fintopo2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& l1_fintopo2(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_2_0_fintopo2(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& A = D
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
=> ( r2_hidden(E,k6_fintopo2(B,D))
=> ( ~ r1_xboole_0(E,C)
& ~ r1_xboole_0(E,k3_subset_1(u1_struct_0(B),C)) ) ) ) ) ) ) ).
fof(fraenkel_a_2_1_fintopo2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& l1_fintopo2(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_2_1_fintopo2(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& A = D
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
=> ~ ( r2_hidden(E,k6_fintopo2(B,D))
& r1_xboole_0(E,C) ) ) ) ) ) ).
fof(fraenkel_a_2_2_fintopo2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& l1_fintopo2(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_2_2_fintopo2(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& A = D
& ? [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
& r2_hidden(E,k6_fintopo2(B,D))
& r1_tarski(E,C) ) ) ) ) ).
fof(fraenkel_a_2_3_fintopo2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& l1_fintopo2(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_2_3_fintopo2(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& A = D
& r2_hidden(D,C)
& ? [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
& r2_hidden(E,k6_fintopo2(B,D))
& r1_xboole_0(k6_subset_1(u1_struct_0(B),E,k1_struct_0(B,D)),C) ) ) ) ) ).
%------------------------------------------------------------------------------