SET007 Axioms: SET007+812.ax
%------------------------------------------------------------------------------
% File : SET007+812 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Some Set Series in Finite Topological Spaces.
% Version : [Urb08] axioms.
% English : Fundamental Concepts for Image Processing
% 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 : fintopo3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 71 ( 0 unt; 0 def)
% Number of atoms : 401 ( 57 equ)
% Maximal formula atoms : 12 ( 5 avg)
% Number of connectives : 405 ( 75 ~; 0 |; 116 &)
% ( 8 <=>; 206 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 8 avg)
% Maximal term depth : 6 ( 2 avg)
% Number of predicates : 12 ( 11 usr; 0 prp; 1-3 aty)
% Number of functors : 29 ( 29 usr; 5 con; 0-4 aty)
% Number of variables : 209 ( 208 !; 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(t1_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_fin_topo(A)
=> r1_tarski(B,k11_fin_topo(A,B)) ) ) ) ).
fof(t2_fintopo3,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,u1_struct_0(A))
=> ( r2_hidden(C,k1_fintopo3(A,B))
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( r2_hidden(D,k3_subset_1(u1_struct_0(A),B))
& r2_hidden(C,k1_fin_topo(A,D)) ) ) ) ) ) ) ).
fof(t3_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_fin_topo(A)
=> r1_tarski(k1_fintopo3(A,B),B) ) ) ) ).
fof(t4_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k1_fintopo3(A,B) = k3_subset_1(u1_struct_0(A),k11_fin_topo(A,k3_subset_1(u1_struct_0(A),B))) ) ) ).
fof(t5_fintopo3,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(k11_fin_topo(A,B),k11_fin_topo(A,C)) ) ) ) ) ).
fof(t6_fintopo3,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(k1_fintopo3(A,B),k1_fintopo3(A,C)) ) ) ) ) ).
fof(t7_fintopo3,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(k8_fin_topo(A,k5_subset_1(u1_struct_0(A),B,C)),k5_subset_1(u1_struct_0(A),k8_fin_topo(A,B),k8_fin_topo(A,C))) ) ) ) ).
fof(t8_fintopo3,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)))
=> k8_fin_topo(A,k4_subset_1(u1_struct_0(A),B,C)) = k4_subset_1(u1_struct_0(A),k8_fin_topo(A,B),k8_fin_topo(A,C)) ) ) ) ).
fof(t9_fintopo3,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(k4_subset_1(u1_struct_0(A),k7_fin_topo(A,B),k7_fin_topo(A,C)),k7_fin_topo(A,k4_subset_1(u1_struct_0(A),B,C))) ) ) ) ).
fof(t10_fintopo3,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)))
=> k5_subset_1(u1_struct_0(A),k7_fin_topo(A,B),k7_fin_topo(A,C)) = k7_fin_topo(A,k5_subset_1(u1_struct_0(A),B,C)) ) ) ) ).
fof(t11_fintopo3,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)))
=> k4_subset_1(u1_struct_0(A),k11_fin_topo(A,B),k11_fin_topo(A,C)) = k11_fin_topo(A,k4_subset_1(u1_struct_0(A),B,C)) ) ) ) ).
fof(t12_fintopo3,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)))
=> k5_subset_1(u1_struct_0(A),k1_fintopo3(A,B),k1_fintopo3(A,C)) = k1_fintopo3(A,k5_subset_1(u1_struct_0(A),B,C)) ) ) ) ).
fof(d2_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_zfmisc_1(u1_struct_0(A)))
& m2_relset_1(C,k5_numbers,k1_zfmisc_1(u1_struct_0(A))) )
=> ( C = k2_fintopo3(A,B)
<=> ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( E = k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),C,D)
=> k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),C,k1_nat_1(D,np__1)) = k8_fin_topo(A,E) ) ) )
& k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),C,np__0) = B ) ) ) ) ) ).
fof(d3_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k3_fintopo3(A,B,C) = k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),k2_fintopo3(A,B),C) ) ) ) ).
fof(d4_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_zfmisc_1(u1_struct_0(A)))
& m2_relset_1(C,k5_numbers,k1_zfmisc_1(u1_struct_0(A))) )
=> ( C = k4_fintopo3(A,B)
<=> ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( E = k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),C,D)
=> k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),C,k1_nat_1(D,np__1)) = k7_fin_topo(A,E) ) ) )
& k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),C,np__0) = B ) ) ) ) ) ).
fof(d5_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k5_fintopo3(A,B,C) = k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),k4_fintopo3(A,B),C) ) ) ) ).
fof(t13_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k3_fintopo3(A,B,k1_nat_1(C,np__1)) = k8_fin_topo(A,k3_fintopo3(A,B,C)) ) ) ) ).
fof(t14_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k3_fintopo3(A,B,np__0) = B ) ) ).
fof(t15_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k3_fintopo3(A,B,np__1) = k8_fin_topo(A,B) ) ) ).
fof(t16_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k3_fintopo3(A,B,np__2) = k8_fin_topo(A,k8_fin_topo(A,B)) ) ) ).
fof(t17_fintopo3,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)))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k3_fintopo3(A,k4_subset_1(u1_struct_0(A),B,C),D) = k4_subset_1(u1_struct_0(A),k3_fintopo3(A,B,D),k3_fintopo3(A,C,D)) ) ) ) ) ).
fof(t18_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k5_fintopo3(A,B,k1_nat_1(C,np__1)) = k7_fin_topo(A,k5_fintopo3(A,B,C)) ) ) ) ).
fof(t19_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k5_fintopo3(A,B,np__0) = B ) ) ).
fof(t20_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k5_fintopo3(A,B,np__1) = k7_fin_topo(A,B) ) ) ).
fof(t21_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k5_fintopo3(A,B,np__2) = k7_fin_topo(A,k7_fin_topo(A,B)) ) ) ).
fof(t22_fintopo3,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)))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k5_fintopo3(A,k5_subset_1(u1_struct_0(A),B,C),D) = k5_subset_1(u1_struct_0(A),k5_fintopo3(A,B,D),k5_fintopo3(A,C,D)) ) ) ) ) ).
fof(t23_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_fin_topo(A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r1_tarski(B,k3_fintopo3(A,B,C)) ) ) ) ) ).
fof(t24_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_fin_topo(A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r1_tarski(k5_fintopo3(A,B,C),B) ) ) ) ) ).
fof(t25_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_fin_topo(A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r1_tarski(k3_fintopo3(A,B,C),k3_fintopo3(A,B,k1_nat_1(C,np__1))) ) ) ) ) ).
fof(t26_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_fin_topo(A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r1_tarski(k5_fintopo3(A,B,k1_nat_1(C,np__1)),k5_fintopo3(A,B,C)) ) ) ) ) ).
fof(t27_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k3_subset_1(u1_struct_0(A),k5_fintopo3(A,k3_subset_1(u1_struct_0(A),B),C)) = k3_fintopo3(A,B,C) ) ) ) ).
fof(t28_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k3_subset_1(u1_struct_0(A),k3_fintopo3(A,k3_subset_1(u1_struct_0(A),B),C)) = k5_fintopo3(A,B,C) ) ) ) ).
fof(t29_fintopo3,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)))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k4_subset_1(u1_struct_0(A),k3_fintopo3(A,B,D),k3_fintopo3(A,C,D)) = k3_subset_1(u1_struct_0(A),k5_fintopo3(A,k3_subset_1(u1_struct_0(A),k4_subset_1(u1_struct_0(A),B,C)),D)) ) ) ) ) ).
fof(t30_fintopo3,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)))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k5_subset_1(u1_struct_0(A),k5_fintopo3(A,B,D),k5_fintopo3(A,C,D)) = k3_subset_1(u1_struct_0(A),k3_fintopo3(A,k3_subset_1(u1_struct_0(A),k5_subset_1(u1_struct_0(A),B,C)),D)) ) ) ) ) ).
fof(d6_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_zfmisc_1(u1_struct_0(A)))
& m2_relset_1(C,k5_numbers,k1_zfmisc_1(u1_struct_0(A))) )
=> ( C = k6_fintopo3(A,B)
<=> ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( E = k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),C,D)
=> k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),C,k1_nat_1(D,np__1)) = k11_fin_topo(A,E) ) ) )
& k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),C,np__0) = B ) ) ) ) ) ).
fof(d7_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k7_fintopo3(A,B,C) = k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),k6_fintopo3(A,B),C) ) ) ) ).
fof(d8_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_zfmisc_1(u1_struct_0(A)))
& m2_relset_1(C,k5_numbers,k1_zfmisc_1(u1_struct_0(A))) )
=> ( C = k8_fintopo3(A,B)
<=> ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( E = k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),C,D)
=> k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),C,k1_nat_1(D,np__1)) = k1_fintopo3(A,E) ) ) )
& k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),C,np__0) = B ) ) ) ) ) ).
fof(d9_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k9_fintopo3(A,B,C) = k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),k8_fintopo3(A,B),C) ) ) ) ).
fof(t31_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k7_fintopo3(A,B,k1_nat_1(C,np__1)) = k11_fin_topo(A,k7_fintopo3(A,B,C)) ) ) ) ).
fof(t32_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k7_fintopo3(A,B,np__0) = B ) ) ).
fof(t33_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k7_fintopo3(A,B,np__1) = k11_fin_topo(A,B) ) ) ).
fof(t34_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k7_fintopo3(A,B,np__2) = k11_fin_topo(A,k11_fin_topo(A,B)) ) ) ).
fof(t35_fintopo3,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)))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k7_fintopo3(A,k4_subset_1(u1_struct_0(A),B,C),D) = k4_subset_1(u1_struct_0(A),k7_fintopo3(A,B,D),k7_fintopo3(A,C,D)) ) ) ) ) ).
fof(t36_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_fin_topo(A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r1_tarski(B,k7_fintopo3(A,B,C)) ) ) ) ) ).
fof(t37_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_fin_topo(A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r1_tarski(k7_fintopo3(A,B,C),k7_fintopo3(A,B,k1_nat_1(C,np__1))) ) ) ) ) ).
fof(t38_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k9_fintopo3(A,B,k1_nat_1(C,np__1)) = k1_fintopo3(A,k9_fintopo3(A,B,C)) ) ) ) ).
fof(t39_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k9_fintopo3(A,B,np__0) = B ) ) ).
fof(t40_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k9_fintopo3(A,B,np__1) = k1_fintopo3(A,B) ) ) ).
fof(t41_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k9_fintopo3(A,B,np__2) = k1_fintopo3(A,k1_fintopo3(A,B)) ) ) ).
fof(t42_fintopo3,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)))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k9_fintopo3(A,k5_subset_1(u1_struct_0(A),B,C),D) = k5_subset_1(u1_struct_0(A),k9_fintopo3(A,B,D),k9_fintopo3(A,C,D)) ) ) ) ) ).
fof(t43_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_fin_topo(A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r1_tarski(k9_fintopo3(A,B,C),B) ) ) ) ) ).
fof(t44_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_fin_topo(A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r1_tarski(k9_fintopo3(A,B,k1_nat_1(C,np__1)),k9_fintopo3(A,B,C)) ) ) ) ) ).
fof(t45_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k9_fintopo3(A,B,C) = k3_subset_1(u1_struct_0(A),k7_fintopo3(A,k3_subset_1(u1_struct_0(A),B),C)) ) ) ) ).
fof(t46_fintopo3,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)))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k5_subset_1(u1_struct_0(A),k9_fintopo3(A,B,D),k9_fintopo3(A,C,D)) = k3_subset_1(u1_struct_0(A),k7_fintopo3(A,k3_subset_1(u1_struct_0(A),k5_subset_1(u1_struct_0(A),B,C)),D)) ) ) ) ) ).
fof(d10_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k10_fintopo3(A,B,C) = k7_fintopo3(A,k1_fin_topo(A,C),B) ) ) ) ).
fof(t47_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k10_fintopo3(A,np__0,B) = k1_fin_topo(A,B) ) ) ).
fof(t48_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k10_fintopo3(A,k1_nat_1(C,np__1),B) = k11_fin_topo(A,k10_fintopo3(A,C,B)) ) ) ) ).
fof(d11_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_fin_topo(B) )
=> ( r1_fintopo3(A,B)
<=> ( u1_struct_0(A) = u1_struct_0(B)
& ! [C,D] :
( ( r2_hidden(C,u1_struct_0(A))
& r2_hidden(D,u1_struct_0(B)) )
=> ( r2_hidden(D,k1_funct_1(u1_fin_topo(A),C))
<=> r2_hidden(C,k1_funct_1(u1_fin_topo(B),D)) ) ) ) ) ) ) ).
fof(symmetry_r1_fintopo3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A)
& ~ v3_struct_0(B)
& l1_fin_topo(B) )
=> ( r1_fintopo3(A,B)
=> r1_fintopo3(B,A) ) ) ).
fof(dt_k1_fintopo3,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(k1_fintopo3(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k2_fintopo3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v1_funct_1(k2_fintopo3(A,B))
& v1_funct_2(k2_fintopo3(A,B),k5_numbers,k1_zfmisc_1(u1_struct_0(A)))
& m2_relset_1(k2_fintopo3(A,B),k5_numbers,k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(dt_k3_fintopo3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k5_numbers) )
=> m1_subset_1(k3_fintopo3(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k4_fintopo3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v1_funct_1(k4_fintopo3(A,B))
& v1_funct_2(k4_fintopo3(A,B),k5_numbers,k1_zfmisc_1(u1_struct_0(A)))
& m2_relset_1(k4_fintopo3(A,B),k5_numbers,k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(dt_k5_fintopo3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k5_numbers) )
=> m1_subset_1(k5_fintopo3(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k6_fintopo3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v1_funct_1(k6_fintopo3(A,B))
& v1_funct_2(k6_fintopo3(A,B),k5_numbers,k1_zfmisc_1(u1_struct_0(A)))
& m2_relset_1(k6_fintopo3(A,B),k5_numbers,k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(dt_k7_fintopo3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k5_numbers) )
=> m1_subset_1(k7_fintopo3(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k8_fintopo3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v1_funct_1(k8_fintopo3(A,B))
& v1_funct_2(k8_fintopo3(A,B),k5_numbers,k1_zfmisc_1(u1_struct_0(A)))
& m2_relset_1(k8_fintopo3(A,B),k5_numbers,k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(dt_k9_fintopo3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k5_numbers) )
=> m1_subset_1(k9_fintopo3(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k10_fintopo3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k10_fintopo3(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(d1_fintopo3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k1_fintopo3(A,B) = a_2_0_fintopo3(A,B) ) ) ).
fof(fraenkel_a_2_0_fintopo3,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_fintopo3(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,k3_subset_1(u1_struct_0(B),C))
& r2_hidden(D,k1_fin_topo(B,E)) ) ) ) ) ) ).
%------------------------------------------------------------------------------