SET007 Axioms: SET007+405.ax
%------------------------------------------------------------------------------
% File : SET007+405 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Maximal Anti-Discrete Subspaces of 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 : tex_4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 160 ( 2 unt; 0 def)
% Number of atoms : 1097 ( 72 equ)
% Maximal formula atoms : 16 ( 6 avg)
% Number of connectives : 1199 ( 262 ~; 10 |; 486 &)
% ( 38 <=>; 403 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 8 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 35 ( 33 usr; 1 prp; 0-3 aty)
% Number of functors : 34 ( 34 usr; 1 con; 0-3 aty)
% Number of variables : 408 ( 394 !; 14 ?)
% 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_tex_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v1_xboole_0(k6_pre_topc(A,B))
& v4_pre_topc(k6_pre_topc(A,B),A) ) ) ).
fof(fc2_tex_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v1_xboole_0(k6_pre_topc(A,B))
& v3_pre_topc(k6_pre_topc(A,B),A)
& v4_pre_topc(k6_pre_topc(A,B),A)
& v2_tops_1(k6_pre_topc(A,B),A)
& v3_tops_1(k6_pre_topc(A,B),A)
& v1_tex_2(k6_pre_topc(A,B),k1_zfmisc_1(u1_struct_0(A)))
& v1_membered(k6_pre_topc(A,B))
& v2_membered(k6_pre_topc(A,B))
& v3_membered(k6_pre_topc(A,B))
& v4_membered(k6_pre_topc(A,B))
& v5_membered(k6_pre_topc(A,B)) ) ) ).
fof(fc3_tex_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v1_tex_2(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v1_xboole_0(k6_pre_topc(A,B))
& v4_pre_topc(k6_pre_topc(A,B),A)
& ~ v1_tex_2(k6_pre_topc(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(fc4_tex_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v1_xboole_0(B)
& ~ v1_realset1(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v1_xboole_0(k6_pre_topc(A,B))
& ~ v1_realset1(k6_pre_topc(A,B))
& v4_pre_topc(k6_pre_topc(A,B),A) ) ) ).
fof(fc5_tex_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v1_tex_2(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v1_xboole_0(k1_tops_1(A,B))
& v3_pre_topc(k1_tops_1(A,B),A)
& ~ v1_tex_2(k1_tops_1(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(fc6_tex_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v1_tex_2(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v3_pre_topc(k1_tops_1(A,B),A)
& v1_tex_2(k1_tops_1(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(fc7_tex_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v1_xboole_0(k1_tops_1(A,B))
& v3_pre_topc(k1_tops_1(A,B),A)
& v4_pre_topc(k1_tops_1(A,B),A)
& v2_tops_1(k1_tops_1(A,B),A)
& v3_tops_1(k1_tops_1(A,B),A)
& v1_tex_2(k1_tops_1(A,B),k1_zfmisc_1(u1_struct_0(A)))
& v1_membered(k1_tops_1(A,B))
& v2_membered(k1_tops_1(A,B))
& v3_membered(k1_tops_1(A,B))
& v4_membered(k1_tops_1(A,B))
& v5_membered(k1_tops_1(A,B)) ) ) ).
fof(fc8_tex_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ~ v1_xboole_0(k1_tex_4(A,B)) ) ).
fof(fc9_tex_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ~ v1_xboole_0(k2_tex_4(A,B)) ) ).
fof(fc10_tex_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ v1_xboole_0(k3_tex_4(A,B)) ) ).
fof(fc11_tex_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A)
& v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v1_xboole_0(k3_tex_4(A,B))
& v2_tops_1(k3_tex_4(A,B),A)
& v1_tex_2(k3_tex_4(A,B),k1_zfmisc_1(u1_struct_0(A)))
& v1_membered(k3_tex_4(A,B))
& v2_membered(k3_tex_4(A,B))
& v3_membered(k3_tex_4(A,B))
& v4_membered(k3_tex_4(A,B))
& v5_membered(k3_tex_4(A,B)) ) ) ).
fof(fc12_tex_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A)
& ~ v1_tex_2(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v1_xboole_0(k3_tex_4(A,B))
& ~ v1_tex_2(k3_tex_4(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(fc13_tex_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& l1_pre_topc(A)
& ~ v1_xboole_0(B)
& ~ v1_realset1(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v1_xboole_0(k3_tex_4(A,B))
& ~ v1_realset1(k3_tex_4(A,B)) ) ) ).
fof(cc1_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( v4_tex_4(B,A)
=> ( v2_pre_topc(B)
& v2_tdlat_3(B)
& v3_tdlat_3(B) ) ) ) ) ).
fof(cc2_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( ~ v2_tdlat_3(B)
=> ~ v4_tex_4(B,A) ) ) ) ).
fof(cc3_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( ( ~ v3_struct_0(B)
& v1_tsep_1(B,A)
& v2_tdlat_3(B) )
=> ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& v2_tdlat_3(B)
& v3_tdlat_3(B)
& v4_tex_4(B,A) ) ) ) ) ).
fof(cc4_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( ( ~ v3_struct_0(B)
& v1_tsep_1(B,A)
& ~ v4_tex_4(B,A) )
=> ( ~ v3_struct_0(B)
& ~ v3_realset2(B)
& v2_pre_topc(B)
& ~ v2_tdlat_3(B)
& ~ v4_tex_4(B,A) ) ) ) ) ).
fof(cc5_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( ( ~ v3_struct_0(B)
& v2_tdlat_3(B)
& ~ v4_tex_4(B,A) )
=> ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& ~ v1_tsep_1(B,A)
& v2_tex_2(B,A) ) ) ) ) ).
fof(cc6_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( ( ~ v3_struct_0(B)
& v1_borsuk_1(B,A)
& v2_tdlat_3(B) )
=> ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& v2_tdlat_3(B)
& v3_tdlat_3(B)
& v4_tex_4(B,A) ) ) ) ) ).
fof(cc7_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( ( ~ v3_struct_0(B)
& v1_borsuk_1(B,A)
& ~ v4_tex_4(B,A) )
=> ( ~ v3_struct_0(B)
& ~ v3_realset2(B)
& v2_pre_topc(B)
& ~ v2_tdlat_3(B)
& ~ v4_tex_4(B,A) ) ) ) ) ).
fof(cc8_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( ( ~ v3_struct_0(B)
& v2_tdlat_3(B)
& ~ v4_tex_4(B,A) )
=> ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& ~ v1_borsuk_1(B,A)
& v2_tex_2(B,A) ) ) ) ) ).
fof(fc14_tex_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( ~ v3_struct_0(k5_tex_4(A,B))
& v1_pre_topc(k5_tex_4(A,B)) ) ) ).
fof(rc1_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_pre_topc(B,A)
& v1_pre_topc(B)
& v2_pre_topc(B)
& v2_tdlat_3(B)
& v3_tdlat_3(B)
& v4_tex_4(B,A) ) ) ).
fof(fc15_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)) )
=> ( ~ v3_struct_0(k5_tex_4(A,B))
& v1_pre_topc(k5_tex_4(A,B))
& v2_pre_topc(k5_tex_4(A,B))
& v2_tdlat_3(k5_tex_4(A,B))
& v3_tdlat_3(k5_tex_4(A,B))
& v4_tex_4(k5_tex_4(A,B),A) ) ) ).
fof(fc16_tex_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v3_struct_0(k6_tex_4(A,B))
& v1_pre_topc(k6_tex_4(A,B)) ) ) ).
fof(rc2_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_pre_topc(B,A)
& v1_pre_topc(B)
& ~ v2_tex_2(B,A) ) ) ).
fof(fc17_tex_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& l1_pre_topc(A)
& ~ v1_xboole_0(B)
& ~ v1_realset1(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v3_struct_0(k6_tex_4(A,B))
& ~ v3_realset2(k6_tex_4(A,B))
& v1_pre_topc(k6_tex_4(A,B)) ) ) ).
fof(fc18_tex_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A)
& ~ v1_xboole_0(B)
& ~ v1_tex_2(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v3_struct_0(k6_tex_4(A,B))
& v1_pre_topc(k6_tex_4(A,B))
& ~ v2_tex_2(k6_tex_4(A,B),A) ) ) ).
fof(fc19_tex_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v3_struct_0(k7_tex_4(A,B))
& v1_pre_topc(k7_tex_4(A,B)) ) ) ).
fof(fc20_tex_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& l1_pre_topc(A)
& ~ v1_xboole_0(B)
& ~ v1_realset1(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v3_struct_0(k7_tex_4(A,B))
& ~ v3_realset2(k7_tex_4(A,B))
& v1_pre_topc(k7_tex_4(A,B)) ) ) ).
fof(fc21_tex_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A)
& ~ v1_xboole_0(B)
& ~ v1_tex_2(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v3_struct_0(k7_tex_4(A,B))
& v1_pre_topc(k7_tex_4(A,B))
& ~ v2_tex_2(k7_tex_4(A,B),A) ) ) ).
fof(t3_tex_4,axiom,
$true ).
fof(d1_tex_4,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_tex_4(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v3_pre_topc(D,A)
& r2_hidden(C,D)
& r2_hidden(C,B) )
=> r1_tarski(B,D) ) ) ) ) ) ) ).
fof(d2_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_tex_4(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v4_pre_topc(D,A)
& r2_hidden(C,D)
& r2_hidden(C,B) )
=> r1_tarski(B,D) ) ) ) ) ) ) ).
fof(d3_tex_4,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_tex_4(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v3_pre_topc(C,A)
& ~ r1_xboole_0(B,C)
& ~ r1_tarski(B,C) ) ) ) ) ) ).
fof(d4_tex_4,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_tex_4(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v4_pre_topc(C,A)
& ~ r1_xboole_0(B,C)
& ~ r1_tarski(B,C) ) ) ) ) ) ).
fof(t5_tex_4,axiom,
$true ).
fof(t6_tex_4,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_tex_4(C,A) )
=> v1_tex_4(D,B) ) ) ) ) ) ).
fof(t7_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,B)
& v1_tex_4(B,A) )
=> v1_tex_4(C,A) ) ) ) ) ).
fof(t8_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> v1_tex_4(k1_struct_0(A,B),A) ) ) ).
fof(t9_tex_4,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) ) ) ).
fof(d5_tex_4,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( v2_tex_4(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(C,B)
=> v1_tex_4(C,A) ) ) ) ) ) ).
fof(t10_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( v2_tex_4(B,A)
=> ( k6_setfam_1(u1_struct_0(A),B) = k1_xboole_0
| v1_tex_4(k5_setfam_1(u1_struct_0(A),B),A) ) ) ) ) ).
fof(t11_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( v2_tex_4(B,A)
=> v1_tex_4(k6_setfam_1(u1_struct_0(A),B),A) ) ) ) ).
fof(t12_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> v2_tex_4(k1_tex_4(A,B),A) ) ) ).
fof(t13_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_struct_0(A,B) = k6_setfam_1(u1_struct_0(A),k1_tex_4(A,B)) ) ) ).
fof(t14_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),k5_setfam_1(u1_struct_0(A),k1_tex_4(A,B))) ) ) ).
fof(t15_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> v1_tex_4(k5_setfam_1(u1_struct_0(A),k1_tex_4(A,B)),A) ) ) ).
fof(d7_tex_4,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v3_tex_4(B,A)
<=> ( v1_tex_4(B,A)
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_tex_4(C,A)
& r1_tarski(B,C) )
=> B = C ) ) ) ) ) ) ).
fof(t16_tex_4,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
& v3_tex_4(C,A) )
=> v3_tex_4(D,B) ) ) ) ) ) ).
fof(t17_tex_4,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))) )
=> ~ v3_tex_4(B,A) ) ) ).
fof(t18_tex_4,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)
& v3_pre_topc(B,A) )
=> v3_tex_4(B,A) ) ) ) ).
fof(t19_tex_4,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)
& v4_pre_topc(B,A) )
=> v3_tex_4(B,A) ) ) ) ).
fof(d8_tex_4,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k2_tex_4(A,B) = k5_setfam_1(u1_struct_0(A),k1_tex_4(A,B)) ) ) ).
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(t21_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,u1_struct_0(A))
=> ( ( v1_tex_4(B,A)
& r2_hidden(C,B) )
=> r1_tarski(B,k2_tex_4(A,C)) ) ) ) ) ).
fof(t22_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> v3_tex_4(k2_tex_4(A,B),A) ) ) ).
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(t24_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))
=> ( r1_subset_1(k2_tex_4(A,B),k2_tex_4(A,C))
| k2_tex_4(A,B) = k2_tex_4(A,C) ) ) ) ) ).
fof(t25_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,u1_struct_0(A))
=> ( ( v4_pre_topc(B,A)
& r2_hidden(C,B) )
=> r1_tarski(k2_tex_4(A,C),B) ) ) ) ) ).
fof(t26_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,u1_struct_0(A))
=> ( ( v3_pre_topc(B,A)
& r2_hidden(C,B) )
=> r1_tarski(k2_tex_4(A,C),B) ) ) ) ) ).
fof(d9_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v3_tex_4(B,A)
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& r2_hidden(C,B)
& B = k2_tex_4(A,C) ) ) ) ) ).
fof(t29_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,u1_struct_0(A))
=> ( ( r2_hidden(C,B)
& v3_tex_4(B,A) )
=> B = k2_tex_4(A,C) ) ) ) ) ).
fof(d10_tex_4,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))) )
=> ( v3_tex_4(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(C,B)
=> B = k2_tex_4(A,C) ) ) ) ) ) ).
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(t31_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,u1_struct_0(A))
=> ~ ( ~ r1_xboole_0(k2_tex_4(A,C),k3_tex_4(A,B))
& r1_xboole_0(k2_tex_4(A,C),B) ) ) ) ) ).
fof(t32_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,u1_struct_0(A))
=> ( ~ r1_xboole_0(k2_tex_4(A,C),k3_tex_4(A,B))
=> r1_tarski(k2_tex_4(A,C),k3_tex_4(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(t34_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> r1_tarski(B,k3_tex_4(A,B)) ) ) ).
fof(t35_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k3_tex_4(A,B) = k3_tex_4(A,k3_tex_4(A,B)) ) ) ).
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(t38_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)))
=> k3_tex_4(A,k4_subset_1(u1_struct_0(A),B,C)) = k4_subset_1(u1_struct_0(A),k3_tex_4(A,B),k3_tex_4(A,C)) ) ) ) ).
fof(t39_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(k3_tex_4(A,k5_subset_1(u1_struct_0(A),B,C)),k5_subset_1(u1_struct_0(A),k3_tex_4(A,B),k3_tex_4(A,C))) ) ) ) ).
fof(t40_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)))
=> ( ( v3_pre_topc(B,A)
& r1_tarski(C,B) )
=> r1_tarski(k3_tex_4(A,C),B) ) ) ) ) ).
fof(t42_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)))
=> ( ( v4_pre_topc(B,A)
& r1_tarski(C,B) )
=> r1_tarski(k3_tex_4(A,C),B) ) ) ) ) ).
fof(d12_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)))
=> ( v1_tex_4(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(C,B)
=> r1_tarski(B,k6_pre_topc(A,k1_struct_0(A,C))) ) ) ) ) ) ).
fof(d13_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)))
=> ( v1_tex_4(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(C,B)
=> k6_pre_topc(A,B) = k6_pre_topc(A,k1_struct_0(A,C)) ) ) ) ) ) ).
fof(d14_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)))
=> ( v1_tex_4(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) )
=> k6_pre_topc(A,k1_struct_0(A,C)) = k6_pre_topc(A,k1_struct_0(A,D)) ) ) ) ) ) ) ).
fof(t44_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_tex_4(C,A)
& r1_tarski(k6_pre_topc(A,k1_struct_0(A,B)),C) )
=> C = k6_pre_topc(A,k1_struct_0(A,B)) ) ) ) ) ).
fof(t45_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)))
=> ( ( v1_tex_4(B,A)
& v4_pre_topc(B,A) )
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(C,B)
=> B = k6_pre_topc(A,k1_struct_0(A,C)) ) ) ) ) ) ).
fof(t46_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)))
=> ( v1_tex_4(B,A)
=> ( v3_pre_topc(B,A)
| v2_tops_1(B,A) ) ) ) ) ).
fof(t47_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( k6_pre_topc(A,k1_struct_0(A,B)) = k1_struct_0(A,B)
=> v3_tex_4(k1_struct_0(A,B),A) ) ) ) ).
fof(t50_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> r1_tarski(k2_tex_4(A,B),k6_pre_topc(A,k1_struct_0(A,B))) ) ) ).
fof(t51_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k2_tex_4(A,B) = k2_tex_4(A,C)
<=> k6_pre_topc(A,k1_struct_0(A,B)) = k6_pre_topc(A,k1_struct_0(A,C)) ) ) ) ) ).
fof(t52_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_subset_1(k2_tex_4(A,B),k2_tex_4(A,C))
<=> k6_pre_topc(A,k1_struct_0(A,B)) != k6_pre_topc(A,k1_struct_0(A,C)) ) ) ) ) ).
fof(t55_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_subset_1(k4_tex_4(A,B),k4_tex_4(A,C))
<=> ~ ( ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v3_pre_topc(D,A)
& r1_tarski(k4_tex_4(A,B),D)
& r1_xboole_0(D,k4_tex_4(A,C)) ) )
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v3_pre_topc(D,A)
& r1_xboole_0(D,k4_tex_4(A,B))
& r1_tarski(k4_tex_4(A,C),D) ) ) ) ) ) ) ) ).
fof(t56_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_subset_1(k4_tex_4(A,B),k4_tex_4(A,C))
<=> ~ ( ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v4_pre_topc(D,A)
& r1_tarski(k4_tex_4(A,B),D)
& r1_xboole_0(D,k4_tex_4(A,C)) ) )
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v4_pre_topc(D,A)
& r1_xboole_0(D,k4_tex_4(A,B))
& r1_tarski(k4_tex_4(A,C),D) ) ) ) ) ) ) ) ).
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(t59_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)))
=> k3_tex_4(A,k1_tops_1(A,B)) = k1_tops_1(A,B) ) ) ).
fof(t61_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)))
=> r1_tarski(k3_tex_4(A,B),k6_pre_topc(A,B)) ) ) ).
fof(t62_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)))
=> ( v4_pre_topc(B,A)
=> k3_tex_4(A,B) = B ) ) ) ).
fof(t63_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)))
=> k3_tex_4(A,k6_pre_topc(A,B)) = k6_pre_topc(A,B) ) ) ).
fof(t64_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)))
=> k6_pre_topc(A,k3_tex_4(A,B)) = k6_pre_topc(A,B) ) ) ).
fof(t65_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)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( k3_tex_4(A,B) = k3_tex_4(A,C)
=> k6_pre_topc(A,B) = k6_pre_topc(A,C) ) ) ) ) ).
fof(t66_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)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v4_pre_topc(B,A)
| v4_pre_topc(C,A) )
=> k3_tex_4(A,k5_subset_1(u1_struct_0(A),B,C)) = k5_subset_1(u1_struct_0(A),k3_tex_4(A,B),k3_tex_4(A,C)) ) ) ) ) ).
fof(t67_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)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v3_pre_topc(B,A)
| v3_pre_topc(C,A) )
=> k3_tex_4(A,k5_subset_1(u1_struct_0(A),B,C)) = k5_subset_1(u1_struct_0(A),k3_tex_4(A,B),k3_tex_4(A,C)) ) ) ) ) ).
fof(t68_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( C = u1_struct_0(B)
& v2_tdlat_3(B) )
=> v1_tex_4(C,A) ) ) ) ) ).
fof(t69_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( v2_pre_topc(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( C = u1_struct_0(B)
& v1_tex_4(C,A) )
=> v2_tdlat_3(B) ) ) ) ) ) ).
fof(t70_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ( ! [C] :
( ( v1_tsep_1(C,A)
& m1_pre_topc(C,A) )
=> ( r1_tsep_1(A,B,C)
| m1_pre_topc(B,C) ) )
=> v2_tdlat_3(B) ) ) ) ).
fof(t71_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ( ! [C] :
( ( v1_borsuk_1(C,A)
& m1_pre_topc(C,A) )
=> ( r1_tsep_1(A,B,C)
| m1_pre_topc(B,C) ) )
=> v2_tdlat_3(B) ) ) ) ).
fof(t72_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_tdlat_3(B)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( v1_tsep_1(C,A)
& m1_pre_topc(C,A) )
=> ( r1_tsep_1(A,B,C)
| m1_pre_topc(B,C) ) ) ) ) ).
fof(t73_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_tdlat_3(B)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( v1_borsuk_1(C,A)
& m1_pre_topc(C,A) )
=> ( r1_tsep_1(A,B,C)
| m1_pre_topc(B,C) ) ) ) ) ).
fof(d16_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( v4_tex_4(B,A)
<=> ( v2_tdlat_3(B)
& ! [C] :
( m1_pre_topc(C,A)
=> ( ( v2_tdlat_3(C)
& r1_tarski(u1_struct_0(B),u1_struct_0(C)) )
=> u1_struct_0(B) = u1_struct_0(C) ) ) ) ) ) ) ).
fof(t74_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( C = u1_struct_0(B)
=> ( v4_tex_4(B,A)
<=> v3_tex_4(C,A) ) ) ) ) ) ).
fof(d17_tex_4,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( v1_pre_topc(C)
& m1_pre_topc(C,A) )
=> ( C = k5_tex_4(A,B)
<=> u1_struct_0(C) = k2_tex_4(A,B) ) ) ) ) ).
fof(t75_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> m1_pre_topc(k1_tex_2(A,B),k5_tex_4(A,B)) ) ) ).
fof(t76_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))
=> ( m1_subset_1(C,u1_struct_0(k5_tex_4(A,B)))
<=> g1_pre_topc(u1_struct_0(k5_tex_4(A,C)),u1_pre_topc(k5_tex_4(A,C))) = g1_pre_topc(u1_struct_0(k5_tex_4(A,B)),u1_pre_topc(k5_tex_4(A,B))) ) ) ) ) ).
fof(t77_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))
=> ( r1_subset_1(u1_struct_0(k5_tex_4(A,B)),u1_struct_0(k5_tex_4(A,C)))
| g1_pre_topc(u1_struct_0(k5_tex_4(A,B)),u1_pre_topc(k5_tex_4(A,B))) = g1_pre_topc(u1_struct_0(k5_tex_4(A,C)),u1_pre_topc(k5_tex_4(A,C))) ) ) ) ) ).
fof(t78_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_borsuk_1(B,A)
& m1_pre_topc(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( m1_subset_1(C,u1_struct_0(B))
=> m1_pre_topc(k5_tex_4(A,C),B) ) ) ) ) ).
fof(t79_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_tsep_1(B,A)
& m1_pre_topc(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( m1_subset_1(C,u1_struct_0(B))
=> m1_pre_topc(k5_tex_4(A,C),B) ) ) ) ) ).
fof(t80_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( k6_pre_topc(A,k1_struct_0(A,B)) = k1_struct_0(A,B)
=> v4_tex_4(k1_tex_2(A,B),A) ) ) ) ).
fof(d18_tex_4,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( ( v1_pre_topc(C)
& m1_pre_topc(C,A) )
=> ( C = k6_tex_4(A,B)
<=> u1_struct_0(C) = B ) ) ) ) ).
fof(t81_tex_4,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))) )
=> m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k6_tex_4(A,B)))) ) ) ).
fof(t82_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
=> m1_pre_topc(k6_tex_4(A,C),B) ) ) ) ) ).
fof(d19_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] :
( ( v1_pre_topc(C)
& m1_pre_topc(C,A) )
=> ( C = k7_tex_4(A,B)
<=> u1_struct_0(C) = k3_tex_4(A,B) ) ) ) ) ).
fof(t83_tex_4,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))) )
=> m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k7_tex_4(A,B)))) ) ) ).
fof(t84_tex_4,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))) )
=> m1_pre_topc(k6_tex_4(A,B),k7_tex_4(A,B)) ) ) ).
fof(t85_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> g1_pre_topc(u1_struct_0(k5_tex_4(A,B)),u1_pre_topc(k5_tex_4(A,B))) = g1_pre_topc(u1_struct_0(k7_tex_4(A,k1_struct_0(A,B))),u1_pre_topc(k7_tex_4(A,k1_struct_0(A,B)))) ) ) ).
fof(t86_tex_4,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))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r1_tarski(B,C)
=> m1_pre_topc(k7_tex_4(A,B),k7_tex_4(A,C)) ) ) ) ) ).
fof(t87_tex_4,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))) )
=> g1_pre_topc(u1_struct_0(k7_tex_4(A,B)),u1_pre_topc(k7_tex_4(A,B))) = g1_pre_topc(u1_struct_0(k7_tex_4(A,k3_tex_4(A,B))),u1_pre_topc(k7_tex_4(A,k3_tex_4(A,B)))) ) ) ).
fof(t88_tex_4,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))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k7_tex_4(A,C))))
=> m1_pre_topc(k7_tex_4(A,B),k7_tex_4(A,C)) ) ) ) ) ).
fof(t89_tex_4,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))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k7_tex_4(A,B))))
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k7_tex_4(A,C)))) )
<=> g1_pre_topc(u1_struct_0(k7_tex_4(A,B)),u1_pre_topc(k7_tex_4(A,B))) = g1_pre_topc(u1_struct_0(k7_tex_4(A,C)),u1_pre_topc(k7_tex_4(A,C))) ) ) ) ) ).
fof(t90_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_tsep_1(B,A)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
=> m1_pre_topc(k7_tex_4(A,C),B) ) ) ) ) ).
fof(t91_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_borsuk_1(B,A)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
=> m1_pre_topc(k7_tex_4(A,C),B) ) ) ) ) ).
fof(dt_k1_tex_4,axiom,
! [A,B] :
( ( l1_pre_topc(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_subset_1(k1_tex_4(A,B),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ).
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_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(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(dt_k5_tex_4,axiom,
! [A,B] :
( ( l1_pre_topc(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( v1_pre_topc(k5_tex_4(A,B))
& m1_pre_topc(k5_tex_4(A,B),A) ) ) ).
fof(dt_k6_tex_4,axiom,
! [A,B] :
( ( l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v1_pre_topc(k6_tex_4(A,B))
& m1_pre_topc(k6_tex_4(A,B),A) ) ) ).
fof(dt_k7_tex_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v1_pre_topc(k7_tex_4(A,B))
& m1_pre_topc(k7_tex_4(A,B),A) ) ) ).
fof(t1_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)))
=> k6_pre_topc(A,B) = k1_setfam_1(a_2_0_tex_4(A,B)) ) ) ).
fof(t2_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k6_pre_topc(A,k1_struct_0(A,B)) = k1_setfam_1(a_2_1_tex_4(A,B)) ) ) ).
fof(t4_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)))
=> k1_tops_1(A,B) = k3_tarski(a_2_2_tex_4(A,B)) ) ) ).
fof(d6_tex_4,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_tex_4(A,B) = a_2_3_tex_4(A,B) ) ) ).
fof(t27_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( a_2_4_tex_4(A,B) != k1_xboole_0
=> r1_tarski(k2_tex_4(A,B),k1_setfam_1(a_2_4_tex_4(A,B))) ) ) ) ).
fof(t28_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( a_2_5_tex_4(A,B) != k1_xboole_0
=> r1_tarski(k2_tex_4(A,B),k1_setfam_1(a_2_5_tex_4(A,B))) ) ) ) ).
fof(d11_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k3_tex_4(A,B) = k3_tarski(a_2_6_tex_4(A,B)) ) ) ).
fof(t41_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( a_2_7_tex_4(A,B) != k1_xboole_0
=> r1_tarski(k3_tex_4(A,B),k1_setfam_1(a_2_7_tex_4(A,B))) ) ) ) ).
fof(t43_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( a_2_8_tex_4(A,B) != k1_xboole_0
=> r1_tarski(k3_tex_4(A,B),k1_setfam_1(a_2_8_tex_4(A,B))) ) ) ) ).
fof(t48_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> r1_tarski(k2_tex_4(A,B),k1_setfam_1(a_2_9_tex_4(A,B))) ) ) ).
fof(t49_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> r1_tarski(k2_tex_4(A,B),k1_setfam_1(a_2_1_tex_4(A,B))) ) ) ).
fof(d15_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k4_tex_4(A,B) = k3_xboole_0(k6_pre_topc(A,k1_struct_0(A,B)),k1_setfam_1(a_2_9_tex_4(A,B))) ) ) ).
fof(t53_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_tarski(k6_pre_topc(A,k1_struct_0(A,B)),k6_pre_topc(A,k1_struct_0(A,C)))
<=> r1_tarski(k1_setfam_1(a_2_9_tex_4(A,C)),k1_setfam_1(a_2_9_tex_4(A,B))) ) ) ) ) ).
fof(t54_tex_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_tarski(k6_pre_topc(A,k1_struct_0(A,B)),k6_pre_topc(A,k1_struct_0(A,C)))
<=> r1_tarski(k4_tex_4(A,C),k1_setfam_1(a_2_9_tex_4(A,B))) ) ) ) ) ).
fof(t57_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)))
=> r1_tarski(k3_tex_4(A,B),k1_setfam_1(a_2_10_tex_4(A,B))) ) ) ).
fof(t60_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)))
=> r1_tarski(k3_tex_4(A,B),k1_setfam_1(a_2_0_tex_4(A,B))) ) ) ).
fof(fraenkel_a_2_0_tex_4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_2_0_tex_4(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
& A = D
& v4_pre_topc(D,B)
& r1_tarski(C,D) ) ) ) ).
fof(fraenkel_a_2_1_tex_4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& m1_subset_1(C,u1_struct_0(B)) )
=> ( r2_hidden(A,a_2_1_tex_4(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
& A = D
& v4_pre_topc(D,B)
& r2_hidden(C,D) ) ) ) ).
fof(fraenkel_a_2_2_tex_4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_2_2_tex_4(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
& A = D
& v3_pre_topc(D,B)
& r1_tarski(D,C) ) ) ) ).
fof(fraenkel_a_2_3_tex_4,axiom,
! [A,B,C] :
( ( l1_pre_topc(B)
& m1_subset_1(C,u1_struct_0(B)) )
=> ( r2_hidden(A,a_2_3_tex_4(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
& A = D
& v1_tex_4(D,B)
& r2_hidden(C,D) ) ) ) ).
fof(fraenkel_a_2_4_tex_4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& l1_pre_topc(B)
& m1_subset_1(C,u1_struct_0(B)) )
=> ( r2_hidden(A,a_2_4_tex_4(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
& A = D
& v4_pre_topc(D,B)
& r2_hidden(C,D) ) ) ) ).
fof(fraenkel_a_2_5_tex_4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& l1_pre_topc(B)
& m1_subset_1(C,u1_struct_0(B)) )
=> ( r2_hidden(A,a_2_5_tex_4(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
& A = D
& v3_pre_topc(D,B)
& r2_hidden(C,D) ) ) ) ).
fof(fraenkel_a_2_6_tex_4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& l1_pre_topc(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_2_6_tex_4(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& A = k2_tex_4(B,D)
& r2_hidden(D,C) ) ) ) ).
fof(fraenkel_a_2_7_tex_4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& l1_pre_topc(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_2_7_tex_4(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
& A = D
& v3_pre_topc(D,B)
& r1_tarski(C,D) ) ) ) ).
fof(fraenkel_a_2_8_tex_4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& l1_pre_topc(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_2_8_tex_4(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
& A = D
& v4_pre_topc(D,B)
& r1_tarski(C,D) ) ) ) ).
fof(fraenkel_a_2_9_tex_4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& m1_subset_1(C,u1_struct_0(B)) )
=> ( r2_hidden(A,a_2_9_tex_4(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
& A = D
& v3_pre_topc(D,B)
& r2_hidden(C,D) ) ) ) ).
fof(fraenkel_a_2_10_tex_4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_2_10_tex_4(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
& A = D
& v3_pre_topc(D,B)
& r1_tarski(C,D) ) ) ) ).
%------------------------------------------------------------------------------