TPTP Problem File: TOP030+1.p
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%------------------------------------------------------------------------------
% File : TOP030+1 : TPTP v8.2.0. Released v3.4.0.
% Domain : Topology
% Problem : Maximal Kolmogorov Subspaces of a Topological Space T15
% Version : [Urb08] axioms : Especial.
% English :
% Refs : [Kar96] Karno (1996), Maximal Kolmogorov Subspaces of a Topolo
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : t15_tsp_2 [Urb08]
% Status : Theorem
% Rating : 1.00 v3.4.0
% Syntax : Number of formulae : 136 ( 17 unt; 0 def)
% Number of atoms : 639 ( 22 equ)
% Maximal formula atoms : 15 ( 4 avg)
% Number of connectives : 618 ( 115 ~; 1 |; 288 &)
% ( 15 <=>; 199 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 6 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 40 ( 38 usr; 1 prp; 0-2 aty)
% Number of functors : 10 ( 10 usr; 1 con; 0-3 aty)
% Number of variables : 249 ( 228 !; 21 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Normal version: includes the axioms (which may be theorems from
% other articles) and background that are possibly necessary.
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : The problem encoding is based on set theory.
%------------------------------------------------------------------------------
fof(t15_tsp_2,conjecture,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_tsp_2(B,A)
& m2_tsp_1(B,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( v3_pre_topc(C,A)
<=> ( C = k3_tex_4(A,C)
& ? [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(antisymmetry_r2_hidden,axiom,
! [A,B] :
( r2_hidden(A,B)
=> ~ r2_hidden(B,A) ) ).
fof(cc10_membered,axiom,
! [A] :
( v1_membered(A)
=> ! [B] :
( m1_subset_1(B,A)
=> v1_xcmplx_0(B) ) ) ).
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(cc10_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( ( v2_tex_2(B,A)
& v2_tsp_2(B,A) )
=> ( v2_pre_topc(B)
& ~ v1_borsuk_1(B,A) ) ) ) ) ).
fof(cc11_membered,axiom,
! [A] :
( v2_membered(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ( v1_xcmplx_0(B)
& v1_xreal_0(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_membered,axiom,
! [A] :
( v3_membered(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ( v1_xcmplx_0(B)
& v1_xreal_0(B)
& v1_rat_1(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_membered,axiom,
! [A] :
( v4_membered(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ( v1_xcmplx_0(B)
& v1_xreal_0(B)
& v1_int_1(B)
& v1_rat_1(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(cc14_membered,axiom,
! [A] :
( v5_membered(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ( v1_xcmplx_0(B)
& v4_ordinal2(B)
& v1_xreal_0(B)
& v1_int_1(B)
& v1_rat_1(B) ) ) ) ).
fof(cc15_membered,axiom,
! [A] :
( v1_xboole_0(A)
=> ( v1_membered(A)
& v2_membered(A)
& v3_membered(A)
& v4_membered(A)
& v5_membered(A) ) ) ).
fof(cc16_membered,axiom,
! [A] :
( v1_membered(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> v1_membered(B) ) ) ).
fof(cc17_membered,axiom,
! [A] :
( v2_membered(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( v1_membered(B)
& v2_membered(B) ) ) ) ).
fof(cc18_membered,axiom,
! [A] :
( v3_membered(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( v1_membered(B)
& v2_membered(B)
& v3_membered(B) ) ) ) ).
fof(cc19_membered,axiom,
! [A] :
( v4_membered(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( v1_membered(B)
& v2_membered(B)
& v3_membered(B)
& v4_membered(B) ) ) ) ).
fof(cc1_membered,axiom,
! [A] :
( v5_membered(A)
=> v4_membered(A) ) ).
fof(cc1_pre_topc,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> v2_pre_topc(B) ) ) ).
fof(cc1_tops_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_xboole_0(B)
=> ( v3_pre_topc(B,A)
& v4_pre_topc(B,A) ) ) ) ) ).
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(cc1_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( ( ~ v3_struct_0(B)
& v2_tsp_2(B,A) )
=> ( ~ v3_struct_0(B)
& v2_t_0topsp(B) ) ) ) ) ).
fof(cc20_membered,axiom,
! [A] :
( v5_membered(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( v1_membered(B)
& v2_membered(B)
& v3_membered(B)
& v4_membered(B)
& v5_membered(B) ) ) ) ).
fof(cc2_membered,axiom,
! [A] :
( v4_membered(A)
=> v3_membered(A) ) ).
fof(cc2_tops_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_xboole_0(B)
=> v2_tops_1(B,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(cc2_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( ( ~ v3_struct_0(B)
& ~ v2_t_0topsp(B) )
=> ( ~ v3_struct_0(B)
& ~ v2_tsp_2(B,A) ) ) ) ) ).
fof(cc3_membered,axiom,
! [A] :
( v3_membered(A)
=> v2_membered(A) ) ).
fof(cc3_tops_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_xboole_0(B)
=> v3_tops_1(B,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(cc3_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( v2_tsp_2(B,A)
=> ( v2_pre_topc(B)
& v1_tex_3(B,A) ) ) ) ) ).
fof(cc4_membered,axiom,
! [A] :
( v2_membered(A)
=> v1_membered(A) ) ).
fof(cc4_tops_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v3_tops_1(B,A)
=> v2_tops_1(B,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(cc4_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( ~ v1_tex_3(B,A)
=> ( v2_pre_topc(B)
& ~ v2_tsp_2(B,A) ) ) ) ) ).
fof(cc5_tops_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v4_pre_topc(B,A)
& v2_tops_1(B,A) )
=> ( v2_tops_1(B,A)
& v3_tops_1(B,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(cc5_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( ( v1_tsep_1(B,A)
& v2_tsp_2(B,A) )
=> ( v2_pre_topc(B)
& ~ v2_tex_2(B,A) ) ) ) ) ).
fof(cc6_tops_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v3_pre_topc(B,A)
& v3_tops_1(B,A) )
=> ( v1_xboole_0(B)
& v3_pre_topc(B,A)
& v4_pre_topc(B,A)
& v1_membered(B)
& v2_membered(B)
& v3_membered(B)
& v4_membered(B)
& v5_membered(B)
& v2_tops_1(B,A)
& v3_tops_1(B,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(cc6_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( ( v1_tsep_1(B,A)
& v2_tex_2(B,A) )
=> ( v2_pre_topc(B)
& ~ v2_tsp_2(B,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(cc7_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( ( v2_tex_2(B,A)
& v2_tsp_2(B,A) )
=> ( v2_pre_topc(B)
& ~ v1_tsep_1(B,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(cc8_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( ( v1_borsuk_1(B,A)
& v2_tsp_2(B,A) )
=> ( v2_pre_topc(B)
& ~ v2_tex_2(B,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(cc9_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( ( v2_tex_2(B,A)
& v1_borsuk_1(B,A) )
=> ( v2_pre_topc(B)
& ~ v2_tsp_2(B,A) ) ) ) ) ).
fof(commutativity_k3_xboole_0,axiom,
! [A,B] : k3_xboole_0(A,B) = k3_xboole_0(B,A) ).
fof(commutativity_k5_subset_1,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> k5_subset_1(A,B,C) = k5_subset_1(A,C,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(d3_tarski,axiom,
! [A,B] :
( r1_tarski(A,B)
<=> ! [C] :
( r2_hidden(C,A)
=> r2_hidden(C,B) ) ) ).
fof(d3_xboole_0,axiom,
! [A,B,C] :
( C = k3_xboole_0(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ( r2_hidden(D,A)
& r2_hidden(D,B) ) ) ) ).
fof(d6_tsp_2,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_2(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& r2_hidden(D,B)
& k5_subset_1(u1_struct_0(A),B,k4_tex_4(A,C)) = k1_struct_0(A,D) ) ) ) ) ) ).
fof(dt_k1_struct_0,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_subset_1(k1_struct_0(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k1_tarski,axiom,
$true ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_tex_4,axiom,
! [A,B] :
( ( l1_pre_topc(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_subset_1(k2_tex_4(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k3_tex_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k3_tex_4(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k3_xboole_0,axiom,
$true ).
fof(dt_k4_tex_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( ~ v1_xboole_0(k4_tex_4(A,B))
& m1_subset_1(k4_tex_4(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(dt_k5_subset_1,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> m1_subset_1(k5_subset_1(A,B,C),k1_zfmisc_1(A)) ) ).
fof(dt_l1_pre_topc,axiom,
! [A] :
( l1_pre_topc(A)
=> l1_struct_0(A) ) ).
fof(dt_l1_struct_0,axiom,
$true ).
fof(dt_m1_pre_topc,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_pre_topc(B,A)
=> l1_pre_topc(B) ) ) ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(dt_m1_tsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_tsp_1(B,A)
=> l1_pre_topc(B) ) ) ).
fof(dt_m2_tsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m2_tsp_1(B,A)
=> l1_pre_topc(B) ) ) ).
fof(dt_u1_struct_0,axiom,
$true ).
fof(existence_l1_pre_topc,axiom,
? [A] : l1_pre_topc(A) ).
fof(existence_l1_struct_0,axiom,
? [A] : l1_struct_0(A) ).
fof(existence_m1_pre_topc,axiom,
! [A] :
( l1_pre_topc(A)
=> ? [B] : m1_pre_topc(B,A) ) ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : m1_subset_1(B,A) ).
fof(existence_m1_tsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ? [B] : m1_tsp_1(B,A) ) ).
fof(existence_m2_tsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ? [B] : m2_tsp_1(B,A) ) ).
fof(fc10_membered,axiom,
! [A] :
( v1_int_1(A)
=> ( v1_membered(k1_tarski(A))
& v2_membered(k1_tarski(A))
& v3_membered(k1_tarski(A))
& v4_membered(k1_tarski(A)) ) ) ).
fof(fc11_membered,axiom,
! [A] :
( v4_ordinal2(A)
=> ( v1_membered(k1_tarski(A))
& v2_membered(k1_tarski(A))
& v3_membered(k1_tarski(A))
& v4_membered(k1_tarski(A))
& v5_membered(k1_tarski(A)) ) ) ).
fof(fc1_struct_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ~ v1_xboole_0(u1_struct_0(A)) ) ).
fof(fc1_subset_1,axiom,
! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ).
fof(fc27_membered,axiom,
! [A,B] :
( v1_membered(A)
=> v1_membered(k3_xboole_0(A,B)) ) ).
fof(fc28_membered,axiom,
! [A,B] :
( v1_membered(A)
=> v1_membered(k3_xboole_0(B,A)) ) ).
fof(fc29_membered,axiom,
! [A,B] :
( v2_membered(A)
=> ( v1_membered(k3_xboole_0(A,B))
& v2_membered(k3_xboole_0(A,B)) ) ) ).
fof(fc2_subset_1,axiom,
! [A] : ~ v1_xboole_0(k1_tarski(A)) ).
fof(fc30_membered,axiom,
! [A,B] :
( v2_membered(A)
=> ( v1_membered(k3_xboole_0(B,A))
& v2_membered(k3_xboole_0(B,A)) ) ) ).
fof(fc31_membered,axiom,
! [A,B] :
( v3_membered(A)
=> ( v1_membered(k3_xboole_0(A,B))
& v2_membered(k3_xboole_0(A,B))
& v3_membered(k3_xboole_0(A,B)) ) ) ).
fof(fc32_membered,axiom,
! [A,B] :
( v3_membered(A)
=> ( v1_membered(k3_xboole_0(B,A))
& v2_membered(k3_xboole_0(B,A))
& v3_membered(k3_xboole_0(B,A)) ) ) ).
fof(fc33_membered,axiom,
! [A,B] :
( v4_membered(A)
=> ( v1_membered(k3_xboole_0(A,B))
& v2_membered(k3_xboole_0(A,B))
& v3_membered(k3_xboole_0(A,B))
& v4_membered(k3_xboole_0(A,B)) ) ) ).
fof(fc34_membered,axiom,
! [A,B] :
( v4_membered(A)
=> ( v1_membered(k3_xboole_0(B,A))
& v2_membered(k3_xboole_0(B,A))
& v3_membered(k3_xboole_0(B,A))
& v4_membered(k3_xboole_0(B,A)) ) ) ).
fof(fc35_membered,axiom,
! [A,B] :
( v5_membered(A)
=> ( v1_membered(k3_xboole_0(A,B))
& v2_membered(k3_xboole_0(A,B))
& v3_membered(k3_xboole_0(A,B))
& v4_membered(k3_xboole_0(A,B))
& v5_membered(k3_xboole_0(A,B)) ) ) ).
fof(fc36_membered,axiom,
! [A,B] :
( v5_membered(A)
=> ( v1_membered(k3_xboole_0(B,A))
& v2_membered(k3_xboole_0(B,A))
& v3_membered(k3_xboole_0(B,A))
& v4_membered(k3_xboole_0(B,A))
& v5_membered(k3_xboole_0(B,A)) ) ) ).
fof(fc6_membered,axiom,
( v1_xboole_0(k1_xboole_0)
& v1_membered(k1_xboole_0)
& v2_membered(k1_xboole_0)
& v3_membered(k1_xboole_0)
& v4_membered(k1_xboole_0)
& v5_membered(k1_xboole_0) ) ).
fof(fc7_membered,axiom,
! [A] :
( v1_xcmplx_0(A)
=> v1_membered(k1_tarski(A)) ) ).
fof(fc8_membered,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v1_membered(k1_tarski(A))
& v2_membered(k1_tarski(A)) ) ) ).
fof(fc9_membered,axiom,
! [A] :
( v1_rat_1(A)
=> ( v1_membered(k1_tarski(A))
& v2_membered(k1_tarski(A))
& v3_membered(k1_tarski(A)) ) ) ).
fof(idempotence_k3_xboole_0,axiom,
! [A,B] : k3_xboole_0(A,A) = A ).
fof(idempotence_k5_subset_1,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> k5_subset_1(A,B,B) = B ) ).
fof(l19_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m2_tsp_1(B,A)
=> m1_subset_1(u1_struct_0(B),k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(rc1_membered,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_membered(A)
& v2_membered(A)
& v3_membered(A)
& v4_membered(A)
& v5_membered(A) ) ).
fof(rc1_subset_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& ~ v1_xboole_0(B) ) ) ).
fof(rc1_tops_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& v3_pre_topc(B,A) ) ) ).
fof(rc2_subset_1,axiom,
! [A] :
? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& v1_xboole_0(B) ) ).
fof(rc2_tops_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& v3_pre_topc(B,A)
& v4_pre_topc(B,A) ) ) ).
fof(rc3_struct_0,axiom,
? [A] :
( l1_struct_0(A)
& ~ v3_struct_0(A) ) ).
fof(rc3_tops_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_xboole_0(B)
& v3_pre_topc(B,A)
& v4_pre_topc(B,A) ) ) ).
fof(rc4_tops_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& v1_xboole_0(B)
& v1_membered(B)
& v2_membered(B)
& v3_membered(B)
& v4_membered(B)
& v5_membered(B)
& v2_tops_1(B,A) ) ) ).
fof(rc5_struct_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(B) ) ) ).
fof(rc5_tops_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& v1_xboole_0(B)
& v3_pre_topc(B,A)
& v4_pre_topc(B,A)
& v1_membered(B)
& v2_membered(B)
& v3_membered(B)
& v4_membered(B)
& v5_membered(B)
& v2_tops_1(B,A)
& v3_tops_1(B,A) ) ) ).
fof(rc6_pre_topc,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& v4_pre_topc(B,A) ) ) ).
fof(rc7_pre_topc,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_xboole_0(B)
& v4_pre_topc(B,A) ) ) ).
fof(redefinition_k1_struct_0,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> k1_struct_0(A,B) = k1_tarski(B) ) ).
fof(redefinition_k4_tex_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> k4_tex_4(A,B) = k2_tex_4(A,B) ) ).
fof(redefinition_k5_subset_1,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> k5_subset_1(A,B,C) = k3_xboole_0(B,C) ) ).
fof(redefinition_m1_tsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_tsp_1(B,A)
<=> m1_pre_topc(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(reflexivity_r1_tarski,axiom,
! [A,B] : r1_tarski(A,A) ).
fof(t11_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [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_2(C,A)
<=> v2_tsp_2(B,A) ) ) ) ) ) ).
fof(t14_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_tsp_2(B,A)
& m2_tsp_1(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(B)))
=> ( D = C
=> ( v3_pre_topc(D,B)
<=> ( v3_pre_topc(k3_tex_4(A,C),A)
& D = k3_xboole_0(k3_tex_4(A,C),u1_struct_0(B)) ) ) ) ) ) ) ) ).
fof(t17_xboole_1,axiom,
! [A,B] : r1_tarski(k3_xboole_0(A,B),A) ).
fof(t1_subset,axiom,
! [A,B] :
( r2_hidden(A,B)
=> m1_subset_1(A,B) ) ).
fof(t1_xboole_1,axiom,
! [A,B,C] :
( ( r1_tarski(A,B)
& r1_tarski(B,C) )
=> r1_tarski(A,C) ) ).
fof(t20_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> r1_tarski(k1_struct_0(A,B),k2_tex_4(A,B)) ) ) ).
fof(t23_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(C,k2_tex_4(A,B))
<=> k2_tex_4(A,C) = k2_tex_4(A,B) ) ) ) ) ).
fof(t2_boole,axiom,
! [A] : k3_xboole_0(A,k1_xboole_0) = k1_xboole_0 ).
fof(t2_subset,axiom,
! [A,B] :
( m1_subset_1(A,B)
=> ( v1_xboole_0(B)
| r2_hidden(A,B) ) ) ).
fof(t30_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k2_tex_4(A,B) = k3_tex_4(A,k1_struct_0(A,B)) ) ) ).
fof(t33_tex_4,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(B,C)
=> r1_tarski(k3_tex_4(A,B),k3_tex_4(A,C)) ) ) ) ) ).
fof(t36_tex_4,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(B,k3_tex_4(A,C))
=> r1_tarski(k3_tex_4(A,B),k3_tex_4(A,C)) ) ) ) ) ).
fof(t37_tex_4,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,k3_tex_4(A,B))
& r1_tarski(B,k3_tex_4(A,C)) )
<=> k3_tex_4(A,B) = k3_tex_4(A,C) ) ) ) ) ).
fof(t37_zfmisc_1,axiom,
! [A,B] :
( r1_tarski(k1_tarski(A),B)
<=> r2_hidden(A,B) ) ).
fof(t3_subset,axiom,
! [A,B] :
( m1_subset_1(A,k1_zfmisc_1(B))
<=> r1_tarski(A,B) ) ).
fof(t4_subset,axiom,
! [A,B,C] :
( ( r2_hidden(A,B)
& m1_subset_1(B,k1_zfmisc_1(C)) )
=> m1_subset_1(A,C) ) ).
fof(t58_tex_4,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)))
=> ( v3_pre_topc(B,A)
=> k3_tex_4(A,B) = B ) ) ) ).
fof(t5_subset,axiom,
! [A,B,C] :
~ ( r2_hidden(A,B)
& m1_subset_1(B,k1_zfmisc_1(C))
& v1_xboole_0(C) ) ).
fof(t6_boole,axiom,
! [A] :
( v1_xboole_0(A)
=> A = k1_xboole_0 ) ).
fof(t7_boole,axiom,
! [A,B] :
~ ( r2_hidden(A,B)
& v1_xboole_0(B) ) ).
fof(t8_boole,axiom,
! [A,B] :
~ ( v1_xboole_0(A)
& A != B
& v1_xboole_0(B) ) ).
%------------------------------------------------------------------------------