SET007 Axioms: SET007+587.ax
%------------------------------------------------------------------------------
% File : SET007+587 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Properties of the Product of Compact 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 : borsuk_3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 44 ( 0 unt; 0 def)
% Number of atoms : 422 ( 20 equ)
% Maximal formula atoms : 19 ( 9 avg)
% Number of connectives : 441 ( 63 ~; 0 |; 247 &)
% ( 5 <=>; 126 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 10 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 30 ( 29 usr; 0 prp; 1-3 aty)
% Number of functors : 24 ( 24 usr; 0 con; 1-5 aty)
% Number of variables : 140 ( 132 !; 8 ?)
% 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(fc1_borsuk_3,axiom,
! [A,B] :
( v1_xboole_0(B)
=> v1_xboole_0(k2_zfmisc_1(A,B)) ) ).
fof(fc2_borsuk_3,axiom,
! [A,B] :
( v1_xboole_0(A)
=> v1_xboole_0(k2_zfmisc_1(A,B)) ) ).
fof(fc3_borsuk_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ( v1_relat_1(k7_grcat_1(A))
& v1_funct_1(k7_grcat_1(A))
& v2_funct_1(k7_grcat_1(A))
& ~ v1_xboole_0(k7_grcat_1(A))
& v1_funct_2(k7_grcat_1(A),u1_struct_0(A),u1_struct_0(A))
& v5_pre_topc(k7_grcat_1(A),A,A)
& v3_tops_2(k7_grcat_1(A),A,A)
& v1_partfun1(k7_grcat_1(A),u1_struct_0(A),u1_struct_0(A))
& v1_t_0topsp(k7_grcat_1(A),A,A) ) ) ).
fof(fc4_borsuk_3,axiom,
! [A,B] :
( ( l1_pre_topc(A)
& v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v3_struct_0(k3_pre_topc(A,B))
& v1_pre_topc(k3_pre_topc(A,B))
& v2_t_0topsp(k3_pre_topc(A,B)) ) ) ).
fof(rc1_borsuk_3,axiom,
? [A] :
( l1_pre_topc(A)
& v3_struct_0(A)
& v1_pre_topc(A)
& v2_pre_topc(A)
& v2_t_0topsp(A) ) ).
fof(cc1_borsuk_3,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ( v3_struct_0(A)
& v2_pre_topc(A) )
=> ( v2_pre_topc(A)
& v2_compts_1(A) ) ) ) ).
fof(fc5_borsuk_3,axiom,
! [A,B] :
( ( v2_pre_topc(A)
& l1_pre_topc(A)
& v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ( v3_struct_0(k6_borsuk_1(A,B))
& v1_pre_topc(k6_borsuk_1(A,B))
& v2_pre_topc(k6_borsuk_1(A,B))
& v2_t_0topsp(k6_borsuk_1(A,B))
& v2_compts_1(k6_borsuk_1(A,B)) ) ) ).
fof(fc6_borsuk_3,axiom,
! [A,B] :
( ( v2_pre_topc(A)
& v2_compts_1(A)
& l1_pre_topc(A)
& v2_pre_topc(B)
& v2_compts_1(B)
& l1_pre_topc(B) )
=> ( v1_pre_topc(k6_borsuk_1(A,B))
& v2_pre_topc(k6_borsuk_1(A,B))
& v2_compts_1(k6_borsuk_1(A,B)) ) ) ).
fof(rc2_borsuk_3,axiom,
! [A] :
( l1_pre_topc(A)
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& v6_compts_1(B,A) ) ) ).
fof(fc7_borsuk_3,axiom,
! [A,B] :
( ( v2_pre_topc(A)
& l1_pre_topc(A)
& v6_compts_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v1_pre_topc(k3_pre_topc(A,B))
& v2_pre_topc(k3_pre_topc(A,B))
& v2_compts_1(k3_pre_topc(A,B)) ) ) ).
fof(t1_borsuk_3,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> k2_pre_topc(k6_borsuk_1(A,B)) = k7_borsuk_1(A,B,k2_pre_topc(A),k2_pre_topc(B)) ) ) ).
fof(t2_borsuk_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( v1_funct_1(k3_borsuk_1(B,A,C))
& v1_funct_2(k3_borsuk_1(B,A,C),u1_struct_0(B),u1_struct_0(k3_pre_topc(A,k1_struct_0(A,C))))
& v5_pre_topc(k3_borsuk_1(B,A,C),B,k3_pre_topc(A,k1_struct_0(A,C)))
& m2_relset_1(k3_borsuk_1(B,A,C),u1_struct_0(B),u1_struct_0(k3_pre_topc(A,k1_struct_0(A,C)))) ) ) ) ) ).
fof(t3_borsuk_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_pre_topc(C)
& l1_pre_topc(C) )
=> ( ( r1_borsuk_3(A,B)
& r1_borsuk_3(B,C) )
=> r1_borsuk_3(A,C) ) ) ) ) ).
fof(t4_borsuk_3,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ( v3_struct_0(k6_borsuk_1(A,B))
& v3_struct_0(k6_borsuk_1(B,A)) ) ) ) ).
fof(t5_borsuk_3,axiom,
! [A] :
( ( v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> v2_compts_1(A) ) ).
fof(t6_borsuk_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k6_borsuk_1(B,k3_pre_topc(A,k1_struct_0(A,C)))),u1_struct_0(B))
& m2_relset_1(D,u1_struct_0(k6_borsuk_1(B,k3_pre_topc(A,k1_struct_0(A,C)))),u1_struct_0(B)) )
=> ( D = k9_funct_3(u1_struct_0(B),k1_struct_0(A,C))
=> v2_funct_1(D) ) ) ) ) ) ).
fof(t7_borsuk_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k6_borsuk_1(k3_pre_topc(A,k1_struct_0(A,C)),B)),u1_struct_0(B))
& m2_relset_1(D,u1_struct_0(k6_borsuk_1(k3_pre_topc(A,k1_struct_0(A,C)),B)),u1_struct_0(B)) )
=> ( D = k10_funct_3(k1_struct_0(A,C),u1_struct_0(B))
=> v2_funct_1(D) ) ) ) ) ) ).
fof(t8_borsuk_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k6_borsuk_1(B,k3_pre_topc(A,k1_struct_0(A,C)))),u1_struct_0(B))
& m2_relset_1(D,u1_struct_0(k6_borsuk_1(B,k3_pre_topc(A,k1_struct_0(A,C)))),u1_struct_0(B)) )
=> ( D = k9_funct_3(u1_struct_0(B),k1_struct_0(A,C))
=> k2_tops_2(k6_borsuk_1(B,k3_pre_topc(A,k1_struct_0(A,C))),B,D) = k14_funct_3(u1_struct_0(B),u1_struct_0(B),u1_struct_0(A),k7_grcat_1(B),k3_borsuk_1(B,A,C)) ) ) ) ) ) ).
fof(t9_borsuk_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k6_borsuk_1(k3_pre_topc(A,k1_struct_0(A,C)),B)),u1_struct_0(B))
& m2_relset_1(D,u1_struct_0(k6_borsuk_1(k3_pre_topc(A,k1_struct_0(A,C)),B)),u1_struct_0(B)) )
=> ( D = k10_funct_3(k1_struct_0(A,C),u1_struct_0(B))
=> k2_tops_2(k6_borsuk_1(k3_pre_topc(A,k1_struct_0(A,C)),B),B,D) = k14_funct_3(u1_struct_0(B),u1_struct_0(A),u1_struct_0(B),k3_borsuk_1(B,A,C),k7_grcat_1(B)) ) ) ) ) ) ).
fof(t10_borsuk_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k6_borsuk_1(B,k3_pre_topc(A,k1_struct_0(A,C)))),u1_struct_0(B))
& m2_relset_1(D,u1_struct_0(k6_borsuk_1(B,k3_pre_topc(A,k1_struct_0(A,C)))),u1_struct_0(B)) )
=> ( D = k9_funct_3(u1_struct_0(B),k1_struct_0(A,C))
=> v3_tops_2(D,k6_borsuk_1(B,k3_pre_topc(A,k1_struct_0(A,C))),B) ) ) ) ) ) ).
fof(t11_borsuk_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k6_borsuk_1(k3_pre_topc(A,k1_struct_0(A,C)),B)),u1_struct_0(B))
& m2_relset_1(D,u1_struct_0(k6_borsuk_1(k3_pre_topc(A,k1_struct_0(A,C)),B)),u1_struct_0(B)) )
=> ( D = k10_funct_3(k1_struct_0(A,C),u1_struct_0(B))
=> v3_tops_2(D,k6_borsuk_1(k3_pre_topc(A,k1_struct_0(A,C)),B),B) ) ) ) ) ) ).
fof(t15_borsuk_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> r1_borsuk_3(k6_borsuk_1(k3_pre_topc(A,k1_struct_0(A,C)),B),B) ) ) ) ).
fof(t16_borsuk_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ( ( r1_borsuk_3(A,B)
& v2_compts_1(A) )
=> v2_compts_1(B) ) ) ) ).
fof(t17_borsuk_3,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_pre_topc(C,A)
=> m1_pre_topc(k6_borsuk_1(B,C),k6_borsuk_1(B,A)) ) ) ) ).
fof(t18_borsuk_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& v2_compts_1(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(B,A))))
=> ( D = k7_borsuk_1(B,A,k2_pre_topc(B),k1_struct_0(A,C))
=> v6_compts_1(D,k6_borsuk_1(B,A)) ) ) ) ) ) ).
fof(t19_borsuk_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& v2_compts_1(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> v2_compts_1(k6_borsuk_1(B,k3_pre_topc(A,k1_struct_0(A,C)))) ) ) ) ).
fof(t23_borsuk_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v2_compts_1(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& v2_compts_1(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k6_borsuk_1(B,A)))))
=> ~ ( r1_pre_topc(k6_borsuk_1(B,A),C)
& v1_tops_2(C,k6_borsuk_1(B,A))
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k6_borsuk_1(B,A)))))
=> ~ ( r1_tarski(D,C)
& r1_pre_topc(k6_borsuk_1(B,A),D)
& v1_finset_1(D) ) ) ) ) ) ) ).
fof(t24_borsuk_3,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> ( ( v2_compts_1(A)
& v2_compts_1(B) )
=> v2_compts_1(k6_borsuk_1(A,B)) ) ) ) ).
fof(t25_borsuk_3,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_pre_topc(C,A)
=> ! [D] :
( m1_pre_topc(D,B)
=> m1_pre_topc(k6_borsuk_1(C,D),k6_borsuk_1(A,B)) ) ) ) ) ).
fof(t26_borsuk_3,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(B,A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
=> ( C = k7_borsuk_1(B,A,E,D)
=> g1_pre_topc(u1_struct_0(k6_borsuk_1(k3_pre_topc(B,E),k3_pre_topc(A,D))),u1_pre_topc(k6_borsuk_1(k3_pre_topc(B,E),k3_pre_topc(A,D)))) = g1_pre_topc(u1_struct_0(k3_pre_topc(k6_borsuk_1(B,A),C)),u1_pre_topc(k3_pre_topc(k6_borsuk_1(B,A),C))) ) ) ) ) ) ) ).
fof(t27_borsuk_3,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ( ( v6_compts_1(C,A)
& v6_compts_1(D,B) )
=> ( v6_compts_1(k7_borsuk_1(A,B,C,D),k6_borsuk_1(A,B))
& m1_subset_1(k7_borsuk_1(A,B,C,D),k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B)))) ) ) ) ) ) ) ).
fof(symmetry_r1_borsuk_3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& l1_pre_topc(B) )
=> ( r1_borsuk_3(A,B)
=> r1_borsuk_3(B,A) ) ) ).
fof(reflexivity_r1_borsuk_3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& l1_pre_topc(B) )
=> r1_borsuk_3(A,A) ) ).
fof(redefinition_r1_borsuk_3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& l1_pre_topc(B) )
=> ( r1_borsuk_3(A,B)
<=> r1_t_0topsp(A,B) ) ) ).
fof(t12_borsuk_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& v2_compts_1(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v3_pre_topc(C,k6_borsuk_1(A,B))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B)))) )
=> ! [D] :
~ ( r2_hidden(D,a_3_0_borsuk_3(A,B,C))
& ! [E] :
( m1_pboole(E,u1_struct_0(B))
=> ? [F] :
( r2_hidden(F,u1_struct_0(B))
& ! [G] :
( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A)))
=> ! [H] :
( m1_subset_1(H,k1_zfmisc_1(u1_struct_0(B)))
=> ~ ( k1_funct_1(E,F) = k4_tarski(G,H)
& r2_hidden(k4_tarski(D,F),k7_borsuk_1(A,B,G,H))
& v3_pre_topc(G,A)
& v3_pre_topc(H,B)
& r1_tarski(k7_borsuk_1(A,B,G,H),C) ) ) ) ) ) ) ) ) ) ).
fof(t13_borsuk_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& v2_compts_1(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v3_pre_topc(C,k6_borsuk_1(B,A))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(B,A)))) )
=> ! [D] :
~ ( r2_hidden(D,a_3_1_borsuk_3(A,B,C))
& ! [E] :
( ( v3_pre_topc(E,A)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( r2_hidden(D,E)
& r1_tarski(E,a_3_1_borsuk_3(A,B,C)) ) ) ) ) ) ) ).
fof(t14_borsuk_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& v2_compts_1(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v3_pre_topc(C,k6_borsuk_1(B,A))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(B,A)))) )
=> r2_hidden(a_3_1_borsuk_3(A,B,C),u1_pre_topc(A)) ) ) ) ).
fof(t20_borsuk_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v2_compts_1(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& v2_compts_1(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( C = a_2_0_borsuk_3(A,B)
=> ( v1_tops_2(C,A)
& r1_compts_1(A,C,k2_pre_topc(A)) ) ) ) ) ) ).
fof(t21_borsuk_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v2_compts_1(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& v2_compts_1(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k6_borsuk_1(B,A)))))
=> ( ( r1_pre_topc(k6_borsuk_1(B,A),D)
& v1_tops_2(D,k6_borsuk_1(B,A))
& C = a_3_2_borsuk_3(A,B,D) )
=> ( v1_tops_2(C,A)
& r1_pre_topc(A,C) ) ) ) ) ) ) ).
fof(t22_borsuk_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v2_compts_1(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& v2_compts_1(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k6_borsuk_1(B,A)))))
=> ~ ( r1_pre_topc(k6_borsuk_1(B,A),D)
& v1_tops_2(D,k6_borsuk_1(B,A))
& C = a_3_2_borsuk_3(A,B,D)
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ~ ( r1_tarski(E,C)
& v1_finset_1(E)
& r1_pre_topc(A,E) ) ) ) ) ) ) ) ).
fof(fraenkel_a_3_0_borsuk_3,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& ~ v3_struct_0(C)
& v2_pre_topc(C)
& v2_compts_1(C)
& l1_pre_topc(C)
& v3_pre_topc(D,k6_borsuk_1(B,C))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(B,C)))) )
=> ( r2_hidden(A,a_3_0_borsuk_3(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = E
& r1_tarski(k2_zfmisc_1(k1_struct_0(B,E),u1_struct_0(C)),D) ) ) ) ).
fof(fraenkel_a_3_1_borsuk_3,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& ~ v3_struct_0(C)
& v2_pre_topc(C)
& v2_compts_1(C)
& l1_pre_topc(C)
& v3_pre_topc(D,k6_borsuk_1(C,B))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(C,B)))) )
=> ( r2_hidden(A,a_3_1_borsuk_3(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = E
& r1_tarski(k7_borsuk_1(C,B,k2_pre_topc(C),k1_struct_0(B,E)),D) ) ) ) ).
fof(fraenkel_a_2_0_borsuk_3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& v2_compts_1(B)
& l1_pre_topc(B)
& ~ v3_struct_0(C)
& v2_pre_topc(C)
& v2_compts_1(C)
& l1_pre_topc(C) )
=> ( r2_hidden(A,a_2_0_borsuk_3(B,C))
<=> ? [D] :
( v3_pre_topc(D,B)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
& A = D
& r1_tarski(k7_borsuk_1(C,B,k2_pre_topc(C),D),k5_setfam_1(u1_struct_0(k6_borsuk_1(C,B)),k11_borsuk_1(C,B,k2_pre_topc(k6_borsuk_1(C,B))))) ) ) ) ).
fof(fraenkel_a_3_2_borsuk_3,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& v2_compts_1(B)
& l1_pre_topc(B)
& ~ v3_struct_0(C)
& v2_pre_topc(C)
& v2_compts_1(C)
& l1_pre_topc(C)
& m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k6_borsuk_1(C,B))))) )
=> ( r2_hidden(A,a_3_2_borsuk_3(B,C,D))
<=> ? [E] :
( v3_pre_topc(E,B)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
& A = E
& ? [F] :
( m1_subset_1(F,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k6_borsuk_1(C,B)))))
& r1_tarski(F,D)
& v1_finset_1(F)
& r1_tarski(k7_borsuk_1(C,B,k2_pre_topc(C),E),k5_setfam_1(u1_struct_0(k6_borsuk_1(C,B)),F)) ) ) ) ) ).
%------------------------------------------------------------------------------