SET007 Axioms: SET007+406.ax
%------------------------------------------------------------------------------
% File : SET007+406 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On Kolmogorov 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 : tsp_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 63 ( 2 unt; 0 def)
% Number of atoms : 519 ( 24 equ)
% Maximal formula atoms : 17 ( 8 avg)
% Number of connectives : 630 ( 174 ~; 9 |; 255 &)
% ( 22 <=>; 170 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 10 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 32 ( 30 usr; 1 prp; 0-3 aty)
% Number of functors : 11 ( 11 usr; 0 con; 1-3 aty)
% Number of variables : 160 ( 149 !; 11 ?)
% 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(cc1_tsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ( ~ v3_struct_0(A)
& v3_realset2(A) )
=> ( ~ v3_struct_0(A)
& v2_t_0topsp(A) ) ) ) ).
fof(cc2_tsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ( ~ v3_struct_0(A)
& ~ v2_t_0topsp(A) )
=> ( ~ v3_struct_0(A)
& ~ v3_realset2(A) ) ) ) ).
fof(rc1_tsp_1,axiom,
? [A] :
( l1_pre_topc(A)
& ~ v3_struct_0(A)
& v1_pre_topc(A)
& v2_pre_topc(A)
& v2_t_0topsp(A) ) ).
fof(rc2_tsp_1,axiom,
? [A] :
( l1_pre_topc(A)
& ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_pre_topc(A)
& v2_pre_topc(A)
& ~ v2_t_0topsp(A) ) ).
fof(cc3_tsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v1_tdlat_3(A) )
=> ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v2_t_0topsp(A) ) ) ) ).
fof(cc4_tsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& ~ v2_t_0topsp(A) )
=> ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_pre_topc(A)
& ~ v1_tdlat_3(A) ) ) ) ).
fof(cc5_tsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_pre_topc(A)
& v2_tdlat_3(A) )
=> ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_pre_topc(A)
& ~ v2_t_0topsp(A) ) ) ) ).
fof(cc6_tsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v2_tdlat_3(A)
& v2_t_0topsp(A) )
=> ( ~ v3_struct_0(A)
& v3_realset2(A)
& v2_pre_topc(A)
& v1_tdlat_3(A)
& v2_tdlat_3(A)
& v3_tdlat_3(A)
& v4_tdlat_3(A)
& v5_tdlat_3(A)
& v2_t_0topsp(A) ) ) ) ).
fof(cc7_tsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_pre_topc(A)
& v2_t_0topsp(A) )
=> ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_pre_topc(A)
& ~ v2_tdlat_3(A) ) ) ) ).
fof(cc8_tsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v3_tdlat_3(A)
& v2_t_0topsp(A) )
=> ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v1_tdlat_3(A)
& v3_tdlat_3(A)
& v4_tdlat_3(A)
& v5_tdlat_3(A)
& v2_t_0topsp(A) ) ) ) ).
fof(cc9_tsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& ~ v1_tdlat_3(A)
& v3_tdlat_3(A) )
=> ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_pre_topc(A)
& ~ v1_tdlat_3(A)
& ~ v2_t_0topsp(A) ) ) ) ).
fof(cc10_tsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& ~ v1_tdlat_3(A)
& v2_t_0topsp(A) )
=> ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_pre_topc(A)
& ~ v1_tdlat_3(A)
& ~ v2_tdlat_3(A)
& ~ v3_tdlat_3(A) ) ) ) ).
fof(rc3_tsp_1,axiom,
? [A] :
( l1_pre_topc(A)
& ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_pre_topc(A)
& v2_pre_topc(A)
& ~ v2_tdlat_3(A)
& v2_t_0topsp(A) ) ).
fof(rc4_tsp_1,axiom,
? [A] :
( l1_pre_topc(A)
& ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_pre_topc(A)
& v2_pre_topc(A)
& ~ v1_tdlat_3(A)
& ~ v2_t_0topsp(A) ) ).
fof(rc5_tsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_pre_topc(B,A)
& ~ v3_struct_0(B)
& v1_pre_topc(B)
& v2_t_0topsp(B) ) ) ).
fof(rc6_tsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_pre_topc(B,A)
& ~ v3_struct_0(B)
& v1_pre_topc(B)
& v2_tex_2(B,A)
& v2_t_0topsp(B) ) ) ).
fof(cc11_tsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v2_t_0topsp(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( ~ v3_struct_0(B)
=> ( ~ v3_struct_0(B)
& v2_t_0topsp(B) ) ) ) ) ).
fof(cc12_tsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& ~ v2_t_0topsp(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( ( ~ v3_struct_0(B)
& ~ v2_tex_2(B,A) )
=> ( ~ v3_struct_0(B)
& ~ v3_realset2(B)
& ~ v2_t_0topsp(B) ) ) ) ) ).
fof(cc13_tsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& ~ v2_t_0topsp(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( ( ~ v3_struct_0(B)
& v2_t_0topsp(B) )
=> ( ~ v3_struct_0(B)
& v2_tex_2(B,A) ) ) ) ) ).
fof(rc7_tsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& ~ v2_t_0topsp(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_pre_topc(B,A)
& v1_pre_topc(B)
& ~ v2_t_0topsp(B) ) ) ).
fof(d1_tsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( l1_pre_topc(B)
=> ( m1_tsp_1(B,A)
<=> ( r1_tarski(u1_struct_0(B),u1_struct_0(A))
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
=> ( v3_pre_topc(C,B)
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
& v3_pre_topc(D,A)
& C = k3_xboole_0(D,u1_struct_0(B)) ) ) ) ) ) ) ) ).
fof(t1_tsp_1,axiom,
$true ).
fof(t2_tsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_tsp_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v3_pre_topc(C,A)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ~ ( v3_pre_topc(D,B)
& D = k3_xboole_0(C,u1_struct_0(B)) ) ) ) ) ) ) ).
fof(d2_tsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( l1_pre_topc(B)
=> ( m2_tsp_1(B,A)
<=> ( r1_tarski(u1_struct_0(B),u1_struct_0(A))
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
=> ( v4_pre_topc(C,B)
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
& v4_pre_topc(D,A)
& C = k3_xboole_0(D,u1_struct_0(B)) ) ) ) ) ) ) ) ).
fof(t3_tsp_1,axiom,
$true ).
fof(t4_tsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m2_tsp_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v4_pre_topc(C,A)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ~ ( v4_pre_topc(D,B)
& D = k3_xboole_0(C,u1_struct_0(B)) ) ) ) ) ) ) ).
fof(d3_tsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( v2_t_0topsp(A)
<=> ( v3_struct_0(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)))
=> ~ ( v3_pre_topc(D,A)
& r2_hidden(B,D)
& ~ r2_hidden(C,D) ) )
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v3_pre_topc(D,A)
& ~ r2_hidden(B,D)
& r2_hidden(C,D) ) ) ) ) ) ) ) ) ).
fof(d4_tsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( v2_t_0topsp(A)
<=> ( v3_struct_0(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)))
=> ~ ( v4_pre_topc(D,A)
& r2_hidden(B,D)
& ~ r2_hidden(C,D) ) )
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v4_pre_topc(D,A)
& ~ r2_hidden(B,D)
& r2_hidden(C,D) ) ) ) ) ) ) ) ) ).
fof(d5_tsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v2_t_0topsp(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( B != C
& k6_pre_topc(A,k1_struct_0(A,B)) = k6_pre_topc(A,k1_struct_0(A,C)) ) ) ) ) ) ).
fof(d6_tsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v2_t_0topsp(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( B != C
& r2_hidden(B,k6_pre_topc(A,k1_struct_0(A,C)))
& r2_hidden(C,k6_pre_topc(A,k1_struct_0(A,B))) ) ) ) ) ) ).
fof(d7_tsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v2_t_0topsp(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( B != C
& r2_hidden(B,k6_pre_topc(A,k1_struct_0(A,C)))
& r1_tarski(k6_pre_topc(A,k1_struct_0(A,C)),k6_pre_topc(A,k1_struct_0(A,B))) ) ) ) ) ) ).
fof(d8_tsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_tsp_1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( r2_hidden(C,B)
& r2_hidden(D,B)
& C != D
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v3_pre_topc(E,A)
& r2_hidden(C,E)
& ~ r2_hidden(D,E) ) )
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v3_pre_topc(E,A)
& ~ r2_hidden(C,E)
& r2_hidden(D,E) ) ) ) ) ) ) ) ) ).
fof(d9_tsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_tsp_1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( r2_hidden(C,B)
& r2_hidden(D,B)
& C != D
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v4_pre_topc(E,A)
& r2_hidden(C,E)
& ~ r2_hidden(D,E) ) )
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v4_pre_topc(E,A)
& ~ r2_hidden(C,E)
& r2_hidden(D,E) ) ) ) ) ) ) ) ) ).
fof(t5_tsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [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)))
=> ( ( g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) = g1_pre_topc(u1_struct_0(B),u1_pre_topc(B))
& C = D
& v1_tsp_1(C,A) )
=> v1_tsp_1(D,B) ) ) ) ) ) ).
fof(t6_tsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( B = u1_struct_0(A)
=> ( v1_tsp_1(B,A)
<=> v2_t_0topsp(A) ) ) ) ) ).
fof(t7_tsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(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)))
=> ( ( r1_tarski(C,B)
& v1_tsp_1(B,A) )
=> v1_tsp_1(C,A) ) ) ) ) ).
fof(t8_tsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(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)))
=> ( ( v1_tsp_1(B,A)
| v1_tsp_1(C,A) )
=> v1_tsp_1(k5_subset_1(u1_struct_0(A),B,C),A) ) ) ) ) ).
fof(t9_tsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(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)))
=> ( ( v1_tsp_1(B,A)
& v1_tsp_1(C,A) )
=> ( ( ~ v3_pre_topc(B,A)
& ~ v3_pre_topc(C,A) )
| v1_tsp_1(k4_subset_1(u1_struct_0(A),B,C),A) ) ) ) ) ) ).
fof(t10_tsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(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)))
=> ( ( v1_tsp_1(B,A)
& v1_tsp_1(C,A) )
=> ( ( ~ v4_pre_topc(B,A)
& ~ v4_pre_topc(C,A) )
| v1_tsp_1(k4_subset_1(u1_struct_0(A),B,C),A) ) ) ) ) ) ).
fof(t11_tsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v3_tex_2(B,A)
=> v1_tsp_1(B,A) ) ) ) ).
fof(t12_tsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( v1_tex_4(B,A)
& ~ v1_realset1(B)
& v1_tsp_1(B,A) ) ) ) ).
fof(d10_tsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_tsp_1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( r2_hidden(C,B)
& r2_hidden(D,B)
& C != D
& k6_pre_topc(A,k1_struct_0(A,C)) = k6_pre_topc(A,k1_struct_0(A,D)) ) ) ) ) ) ) ).
fof(d11_tsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_tsp_1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( r2_hidden(C,B)
& r2_hidden(D,B)
& C != D
& r2_hidden(C,k6_pre_topc(A,k1_struct_0(A,D)))
& r2_hidden(D,k6_pre_topc(A,k1_struct_0(A,C))) ) ) ) ) ) ) ).
fof(d12_tsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_tsp_1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( r2_hidden(C,B)
& r2_hidden(D,B)
& C != D
& r2_hidden(C,k6_pre_topc(A,k1_struct_0(A,D)))
& r1_tarski(k6_pre_topc(A,k1_struct_0(A,D)),k6_pre_topc(A,k1_struct_0(A,C))) ) ) ) ) ) ) ).
fof(t13_tsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> v1_tsp_1(B,A) ) ) ).
fof(t14_tsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> v1_tsp_1(k1_struct_0(A,B),A) ) ) ).
fof(d13_tsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m2_tsp_1(B,A)
=> ( v2_t_0topsp(B)
<=> ( v3_struct_0(B)
| ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( m1_subset_1(C,u1_struct_0(B))
& m1_subset_1(D,u1_struct_0(B))
& C != D
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v3_pre_topc(E,A)
& r2_hidden(C,E)
& ~ r2_hidden(D,E) ) )
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v3_pre_topc(E,A)
& ~ r2_hidden(C,E)
& r2_hidden(D,E) ) ) ) ) ) ) ) ) ) ).
fof(d14_tsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m2_tsp_1(B,A)
=> ( v2_t_0topsp(B)
<=> ( v3_struct_0(B)
| ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( m1_subset_1(C,u1_struct_0(B))
& m1_subset_1(D,u1_struct_0(B))
& C != D
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v4_pre_topc(E,A)
& r2_hidden(C,E)
& ~ r2_hidden(D,E) ) )
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v4_pre_topc(E,A)
& ~ r2_hidden(C,E)
& r2_hidden(D,E) ) ) ) ) ) ) ) ) ) ).
fof(t15_tsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m2_tsp_1(B,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( C = u1_struct_0(B)
=> ( v1_tsp_1(C,A)
<=> v2_t_0topsp(B) ) ) ) ) ) ).
fof(t16_tsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m2_tsp_1(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_t_0topsp(C)
& m2_tsp_1(C,A) )
=> ( m2_tsp_1(B,C)
=> v2_t_0topsp(B) ) ) ) ) ).
fof(t17_tsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_t_0topsp(B)
& m2_tsp_1(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m2_tsp_1(C,A) )
=> ( ~ r1_tsep_1(A,B,C)
=> v2_t_0topsp(k2_tsep_1(A,B,C)) ) ) ) ) ).
fof(t18_tsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_t_0topsp(B)
& m2_tsp_1(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_t_0topsp(C)
& m2_tsp_1(C,A) )
=> ( ( v1_tsep_1(B,A)
| v1_tsep_1(C,A) )
=> v2_t_0topsp(k1_tsep_1(A,B,C)) ) ) ) ) ).
fof(t19_tsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_t_0topsp(B)
& m2_tsp_1(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_t_0topsp(C)
& m2_tsp_1(C,A) )
=> ( ( v1_borsuk_1(B,A)
| v1_borsuk_1(C,A) )
=> v2_t_0topsp(k1_tsep_1(A,B,C)) ) ) ) ) ).
fof(t20_tsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( v1_tsp_1(B,A)
& ! [C] :
( ( ~ v3_struct_0(C)
& v1_pre_topc(C)
& v2_t_0topsp(C)
& m2_tsp_1(C,A) )
=> B != u1_struct_0(C) ) ) ) ) ).
fof(t21_tsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& ~ v2_t_0topsp(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( ~ v1_tsp_1(B,A)
& ! [C] :
( ( v1_pre_topc(C)
& ~ v2_t_0topsp(C)
& m2_tsp_1(C,A) )
=> B != u1_struct_0(C) ) ) ) ) ).
fof(dt_m1_tsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_tsp_1(B,A)
=> l1_pre_topc(B) ) ) ).
fof(existence_m1_tsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ? [B] : m1_tsp_1(B,A) ) ).
fof(redefinition_m1_tsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_tsp_1(B,A)
<=> m1_pre_topc(B,A) ) ) ).
fof(dt_m2_tsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m2_tsp_1(B,A)
=> l1_pre_topc(B) ) ) ).
fof(existence_m2_tsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ? [B] : m2_tsp_1(B,A) ) ).
fof(redefinition_m2_tsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m2_tsp_1(B,A)
<=> m1_pre_topc(B,A) ) ) ).
fof(redefinition_v2_tsp_1,axiom,
! [A,B] :
( ( l1_pre_topc(A)
& m1_pre_topc(B,A) )
=> ( v2_tsp_1(B,A)
<=> v2_t_0topsp(B) ) ) ).
fof(redefinition_v3_tsp_1,axiom,
! [A,B] :
( ( l1_pre_topc(A)
& m1_pre_topc(B,A) )
=> ( v3_tsp_1(B,A)
<=> v2_t_0topsp(B) ) ) ).
%------------------------------------------------------------------------------