SET007 Axioms: SET007+371.ax
%------------------------------------------------------------------------------
% File : SET007+371 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On Discrete and Almost Discrete 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_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 77 ( 6 unt; 0 def)
% Number of atoms : 499 ( 64 equ)
% Maximal formula atoms : 13 ( 6 avg)
% Number of connectives : 564 ( 142 ~; 7 |; 238 &)
% ( 13 <=>; 164 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 7 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 24 ( 22 usr; 1 prp; 0-2 aty)
% Number of functors : 22 ( 22 usr; 1 con; 0-3 aty)
% Number of variables : 150 ( 134 !; 16 ?)
% 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_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_struct_0(g1_struct_0(A))
& ~ v3_struct_0(g1_struct_0(A)) ) ) ).
fof(rc1_tex_1,axiom,
? [A] :
( l1_struct_0(A)
& v1_struct_0(A)
& ~ v3_struct_0(A)
& v3_realset2(A) ) ).
fof(rc2_tex_1,axiom,
? [A] :
( l1_struct_0(A)
& v1_struct_0(A)
& ~ v3_struct_0(A)
& ~ v3_realset2(A) ) ).
fof(rc3_tex_1,axiom,
? [A] :
( l1_pre_topc(A)
& ~ v3_struct_0(A)
& v3_realset2(A)
& v1_pre_topc(A) ) ).
fof(rc4_tex_1,axiom,
? [A] :
( l1_pre_topc(A)
& ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_pre_topc(A) ) ).
fof(rc5_tex_1,axiom,
? [A] :
( l1_pre_topc(A)
& ~ v3_struct_0(A)
& v3_realset2(A)
& v1_pre_topc(A)
& v2_pre_topc(A) ) ).
fof(cc1_tex_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ( ~ v3_struct_0(A)
& v3_realset2(A)
& v2_pre_topc(A) )
=> ( ~ v3_struct_0(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) ) ) ) ).
fof(cc2_tex_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v1_tdlat_3(A)
& v2_tdlat_3(A) )
=> ( ~ v3_struct_0(A)
& v3_realset2(A)
& v2_pre_topc(A) ) ) ) ).
fof(rc6_tex_1,axiom,
? [A] :
( l1_pre_topc(A)
& ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_pre_topc(A)
& v2_pre_topc(A) ) ).
fof(cc3_tex_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& ~ v1_tdlat_3(A) )
=> ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_pre_topc(A) ) ) ) ).
fof(cc4_tex_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& ~ v2_tdlat_3(A) )
=> ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_pre_topc(A) ) ) ) ).
fof(fc2_tex_1,axiom,
! [A] : ~ v1_xboole_0(k1_tex_1(A)) ).
fof(fc3_tex_1,axiom,
! [A] :
( v1_pre_topc(k2_tex_1(A))
& v2_pre_topc(k2_tex_1(A))
& v2_tdlat_3(k2_tex_1(A))
& v3_tdlat_3(k2_tex_1(A)) ) ).
fof(fc4_tex_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ v3_struct_0(k2_tex_1(A))
& v1_pre_topc(k2_tex_1(A))
& v2_pre_topc(k2_tex_1(A))
& v2_tdlat_3(k2_tex_1(A))
& v3_tdlat_3(k2_tex_1(A))
& v4_tdlat_3(k2_tex_1(A))
& v5_tdlat_3(k2_tex_1(A)) ) ) ).
fof(fc5_tex_1,axiom,
! [A] :
( v1_pre_topc(k2_pcomps_1(A))
& v2_pre_topc(k2_pcomps_1(A))
& v1_tdlat_3(k2_pcomps_1(A))
& v3_tdlat_3(k2_pcomps_1(A)) ) ).
fof(rc7_tex_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_tdlat_3(A)
& v3_tdlat_3(A)
& v4_tdlat_3(A)
& v5_tdlat_3(A) ) ).
fof(rc8_tex_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_tdlat_3(A)
& v3_tdlat_3(A)
& v4_tdlat_3(A)
& v5_tdlat_3(A) ) ).
fof(fc6_tex_1,axiom,
! [A,B] :
( m1_subset_1(B,A)
=> ( v1_pre_topc(k4_tex_1(A,B))
& v2_pre_topc(k4_tex_1(A,B)) ) ) ).
fof(fc7_tex_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A) )
=> ( ~ v3_struct_0(k4_tex_1(A,B))
& v1_pre_topc(k4_tex_1(A,B))
& v2_pre_topc(k4_tex_1(A,B)) ) ) ).
fof(rc9_tex_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_tdlat_3(A) ) ).
fof(cc5_tex_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& ~ v3_tdlat_3(A) )
=> ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_pre_topc(A)
& ~ v1_tdlat_3(A)
& ~ v2_tdlat_3(A) ) ) ) ).
fof(rc10_tex_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_tdlat_3(A)
& v3_tdlat_3(A)
& v4_tdlat_3(A)
& v5_tdlat_3(A) ) ).
fof(rc11_tex_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_tdlat_3(A)
& ~ v3_tdlat_3(A) ) ).
fof(t1_tex_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)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k6_tmap_1(A,B))))
=> ( ( C = D
& v3_pre_topc(C,A) )
=> v3_pre_topc(D,k6_tmap_1(A,B)) ) ) ) ) ) ).
fof(t2_tex_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)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k6_tmap_1(A,B))))
=> ( ( C = D
& v4_pre_topc(C,A) )
=> v4_pre_topc(D,k6_tmap_1(A,B)) ) ) ) ) ) ).
fof(t3_tex_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)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_tmap_1(A,k3_subset_1(u1_struct_0(A),B)))))
=> ( C = B
=> v4_pre_topc(C,k6_tmap_1(A,k3_subset_1(u1_struct_0(A),B))) ) ) ) ) ).
fof(t4_tex_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)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_tmap_1(A,B))))
=> ( ( C = B
& v1_tops_1(B,A) )
=> ( v1_tops_1(C,k6_tmap_1(A,B))
& v3_pre_topc(C,k6_tmap_1(A,B)) ) ) ) ) ) ).
fof(t5_tex_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)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_tmap_1(A,B))))
=> ( ( r1_tarski(B,C)
& v1_tops_1(B,A) )
=> v1_tops_3(C,k6_tmap_1(A,B)) ) ) ) ) ).
fof(t6_tex_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)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_tmap_1(A,k3_subset_1(u1_struct_0(A),B)))))
=> ( ( C = B
& v2_tops_1(B,A) )
=> ( v2_tops_1(C,k6_tmap_1(A,k3_subset_1(u1_struct_0(A),B)))
& v4_pre_topc(C,k6_tmap_1(A,k3_subset_1(u1_struct_0(A),B))) ) ) ) ) ) ).
fof(t7_tex_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)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_tmap_1(A,k3_subset_1(u1_struct_0(A),B)))))
=> ( ( r1_tarski(C,B)
& v2_tops_1(B,A) )
=> v3_tops_1(C,k6_tmap_1(A,k3_subset_1(u1_struct_0(A),B))) ) ) ) ) ).
fof(d1_tex_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ( v3_realset2(A)
<=> ? [B] :
( m1_subset_1(B,u1_struct_0(A))
& u1_struct_0(A) = k1_struct_0(A,B) ) ) ) ).
fof(t8_tex_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_realset2(A)
& l1_pre_topc(A) )
=> ( v3_tdlat_3(A)
=> ( v1_xboole_0(u1_pre_topc(A))
| v2_pre_topc(A) ) ) ) ).
fof(d2_tex_1,axiom,
! [A] : k1_tex_1(A) = k2_tarski(k1_xboole_0,A) ).
fof(d3_tex_1,axiom,
! [A] : k2_tex_1(A) = g1_pre_topc(A,k1_tex_1(A)) ).
fof(t9_tex_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v2_tdlat_3(A)
& l1_pre_topc(A) )
=> g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) = k2_tex_1(u1_struct_0(A)) ) ).
fof(t10_tex_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) = g1_pre_topc(u1_struct_0(k2_tex_1(u1_struct_0(A))),u1_pre_topc(k2_tex_1(u1_struct_0(A))))
=> v2_tdlat_3(A) ) ) ).
fof(t11_tex_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v2_tdlat_3(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_xboole_0(B)
=> k6_pre_topc(A,B) = k1_xboole_0 )
& ( ~ v1_xboole_0(B)
=> k6_pre_topc(A,B) = u1_struct_0(A) ) ) ) ) ).
fof(t12_tex_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v2_tdlat_3(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( B != u1_struct_0(A)
=> k1_tops_1(A,B) = k1_xboole_0 )
& ( B = u1_struct_0(A)
=> k1_tops_1(A,B) = u1_struct_0(A) ) ) ) ) ).
fof(t13_tex_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)
=> k6_pre_topc(A,B) = u1_struct_0(A) ) )
=> v2_tdlat_3(A) ) ) ).
fof(t14_tex_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)))
=> ( B != u1_struct_0(A)
=> k1_tops_1(A,B) = k1_xboole_0 ) )
=> v2_tdlat_3(A) ) ) ).
fof(t15_tex_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v2_tdlat_3(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( B != k1_xboole_0
=> v1_tops_1(B,A) )
& ( B != u1_struct_0(A)
=> v2_tops_1(B,A) ) ) ) ) ).
fof(t16_tex_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)))
=> ( B != k1_xboole_0
=> v1_tops_1(B,A) ) )
=> v2_tdlat_3(A) ) ) ).
fof(t17_tex_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)))
=> ( B != u1_struct_0(A)
=> v2_tops_1(B,A) ) )
=> v2_tdlat_3(A) ) ) ).
fof(t18_tex_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v1_tdlat_3(A)
& l1_pre_topc(A) )
=> g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) = k2_pcomps_1(u1_struct_0(A)) ) ).
fof(t19_tex_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) = g1_pre_topc(u1_struct_0(k2_pcomps_1(u1_struct_0(A))),u1_pre_topc(k2_pcomps_1(u1_struct_0(A))))
=> v1_tdlat_3(A) ) ) ).
fof(t20_tex_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v1_tdlat_3(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( k6_pre_topc(A,B) = B
& k1_tops_1(A,B) = B ) ) ) ).
fof(t21_tex_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)))
=> k6_pre_topc(A,B) = B )
=> v1_tdlat_3(A) ) ) ).
fof(t22_tex_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)))
=> k1_tops_1(A,B) = B )
=> v1_tdlat_3(A) ) ) ).
fof(t23_tex_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( k2_tex_1(A) = k2_pcomps_1(A)
<=> ? [B] :
( m1_subset_1(B,A)
& A = k1_tarski(B) ) ) ) ).
fof(d4_tex_1,axiom,
$true ).
fof(t24_tex_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k4_tex_1(A,B))))
=> ( ( r1_tarski(k1_tarski(B),C)
=> v4_pre_topc(C,k4_tex_1(A,B)) )
& ( v4_pre_topc(C,k4_tex_1(A,B))
=> ( v1_xboole_0(C)
| r1_tarski(k1_tarski(B),C) ) ) ) ) ) ) ).
fof(t25_tex_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ( ~ v1_xboole_0(k4_xboole_0(A,k1_tarski(B)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k4_tex_1(A,B))))
=> ( ( C = k1_tarski(B)
=> ( v4_pre_topc(C,k4_tex_1(A,B))
& v2_tops_1(C,k4_tex_1(A,B)) ) )
& ( ( v4_pre_topc(C,k4_tex_1(A,B))
& v2_tops_1(C,k4_tex_1(A,B)) )
=> ( v1_xboole_0(C)
| C = k1_tarski(B) ) ) ) ) ) ) ) ).
fof(t26_tex_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k4_tex_1(A,B))))
=> ( ( r1_tarski(C,k4_xboole_0(A,k1_tarski(B)))
=> v3_pre_topc(C,k4_tex_1(A,B)) )
& ( v3_pre_topc(C,k4_tex_1(A,B))
=> ( C = A
| r1_tarski(C,k4_xboole_0(A,k1_tarski(B))) ) ) ) ) ) ) ).
fof(t27_tex_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ( ~ v1_xboole_0(k4_xboole_0(A,k1_tarski(B)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k4_tex_1(A,B))))
=> ( ( C = k4_xboole_0(A,k1_tarski(B))
=> ( v3_pre_topc(C,k4_tex_1(A,B))
& v1_tops_1(C,k4_tex_1(A,B)) ) )
& ( ( v3_pre_topc(C,k4_tex_1(A,B))
& v1_tops_1(C,k4_tex_1(A,B)) )
=> ( C = A
| C = k4_xboole_0(A,k1_tarski(B)) ) ) ) ) ) ) ) ).
fof(t28_tex_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_pre_topc(C)
& l1_pre_topc(C) )
=> ( g1_pre_topc(u1_struct_0(C),u1_pre_topc(C)) = g1_pre_topc(u1_struct_0(k4_tex_1(A,B)),u1_pre_topc(k4_tex_1(A,B)))
<=> ( u1_struct_0(C) = A
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(C)))
=> ( ( r1_tarski(k1_tarski(B),D)
=> v4_pre_topc(D,C) )
& ( v4_pre_topc(D,C)
=> ( v1_xboole_0(D)
| r1_tarski(k1_tarski(B),D) ) ) ) ) ) ) ) ) ) ).
fof(t29_tex_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_pre_topc(C)
& l1_pre_topc(C) )
=> ( g1_pre_topc(u1_struct_0(C),u1_pre_topc(C)) = g1_pre_topc(u1_struct_0(k4_tex_1(A,B)),u1_pre_topc(k4_tex_1(A,B)))
<=> ( u1_struct_0(C) = A
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(C)))
=> ( ( r1_tarski(D,k4_xboole_0(A,k1_tarski(B)))
=> v3_pre_topc(D,C) )
& ( v3_pre_topc(D,C)
=> ( D = A
| r1_tarski(D,k4_xboole_0(A,k1_tarski(B))) ) ) ) ) ) ) ) ) ) ).
fof(t30_tex_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_pre_topc(C)
& l1_pre_topc(C) )
=> ( g1_pre_topc(u1_struct_0(C),u1_pre_topc(C)) = g1_pre_topc(u1_struct_0(k4_tex_1(A,B)),u1_pre_topc(k4_tex_1(A,B)))
<=> ( u1_struct_0(C) = A
& ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(C))) )
=> k6_pre_topc(C,D) = k2_xboole_0(D,k1_tarski(B)) ) ) ) ) ) ) ).
fof(t31_tex_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_pre_topc(C)
& l1_pre_topc(C) )
=> ( g1_pre_topc(u1_struct_0(C),u1_pre_topc(C)) = g1_pre_topc(u1_struct_0(k4_tex_1(A,B)),u1_pre_topc(k4_tex_1(A,B)))
<=> ( u1_struct_0(C) = A
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(C)))
=> ( D != A
=> k1_tops_1(C,D) = k4_xboole_0(D,k1_tarski(B)) ) ) ) ) ) ) ) ).
fof(t32_tex_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ( k4_tex_1(A,B) = k2_tex_1(A)
<=> A = k1_tarski(B) ) ) ) ).
fof(t33_tex_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ( k4_tex_1(A,B) = k2_pcomps_1(A)
<=> A = k1_tarski(B) ) ) ) ).
fof(t34_tex_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k4_tex_1(A,B))))
=> ( C = k1_tarski(B)
=> k2_pcomps_1(A) = k6_tmap_1(k4_tex_1(A,B),C) ) ) ) ) ).
fof(d6_tex_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_tdlat_3(A)
<=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ v2_tops_1(B,A) ) ) ) ).
fof(t35_tex_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_tdlat_3(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( B != u1_struct_0(A)
& v1_tops_1(B,A) ) ) ) ) ).
fof(d7_tex_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v3_tdlat_3(A)
<=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ v3_tops_1(B,A) ) ) ) ).
fof(t36_tex_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v3_tdlat_3(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( B != u1_struct_0(A)
& v1_tops_3(B,A) ) ) ) ) ).
fof(t37_tex_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ~ ( ~ v3_tdlat_3(A)
& ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( v2_tops_1(B,A)
& v4_pre_topc(B,A) ) ) )
& ~ ( ? [B] :
( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& v2_tops_1(B,A)
& v4_pre_topc(B,A) )
& v3_tdlat_3(A) ) ) ) ).
fof(t38_tex_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ~ ( ~ v3_tdlat_3(A)
& ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( B != u1_struct_0(A)
& v1_tops_1(B,A)
& v3_pre_topc(B,A) ) ) )
& ~ ( ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& B != u1_struct_0(A)
& v1_tops_1(B,A)
& v3_pre_topc(B,A) )
& v3_tdlat_3(A) ) ) ) ).
fof(t39_tex_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ( ~ ( ~ v1_xboole_0(k4_xboole_0(A,k1_tarski(B)))
& v3_tdlat_3(k4_tex_1(A,B)) )
& ~ ( ~ v3_tdlat_3(k4_tex_1(A,B))
& v1_xboole_0(k4_xboole_0(A,k1_tarski(B))) ) ) ) ) ).
fof(t40_tex_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))) )
=> ~ ( v2_tops_1(B,A)
& v3_tdlat_3(k6_tmap_1(A,k3_subset_1(u1_struct_0(A),B))) ) ) ) ).
fof(t41_tex_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)))
=> ~ ( B != u1_struct_0(A)
& v1_tops_1(B,A)
& v3_tdlat_3(k6_tmap_1(A,B)) ) ) ) ).
fof(dt_k1_tex_1,axiom,
! [A] : m1_subset_1(k1_tex_1(A),k1_zfmisc_1(k1_zfmisc_1(A))) ).
fof(dt_k2_tex_1,axiom,
! [A] : l1_pre_topc(k2_tex_1(A)) ).
fof(dt_k3_tex_1,axiom,
! [A,B,C] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> m1_subset_1(k3_tex_1(A,B,C),k1_zfmisc_1(k1_zfmisc_1(A))) ) ).
fof(redefinition_k3_tex_1,axiom,
! [A,B,C] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> k3_tex_1(A,B,C) = k4_xboole_0(B,C) ) ).
fof(dt_k4_tex_1,axiom,
! [A,B] :
( m1_subset_1(B,A)
=> l1_pre_topc(k4_tex_1(A,B)) ) ).
fof(d5_tex_1,axiom,
! [A,B] :
( m1_subset_1(B,A)
=> k4_tex_1(A,B) = g1_pre_topc(A,k3_tex_1(A,k1_pcomps_1(A),a_2_0_tex_1(A,B))) ) ).
fof(fraenkel_a_2_0_tex_1,axiom,
! [A,B,C] :
( m1_subset_1(C,B)
=> ( r2_hidden(A,a_2_0_tex_1(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(B))
& A = D
& r2_hidden(C,D)
& D != B ) ) ) ).
%------------------------------------------------------------------------------