Theorem lecldbas 20247

Description: The set of closed intervals forms a closed subbasis for the topology on the extended reals. Since our definition of a basis is in terms of open sets, we express this by showing that the complements of closed intervals form an open subbasis for the topology. (Contributed by Mario Carneiro, 3-Sep-2015.)

Hypothesis

Ref	Expression
lecldbas.1	|- F = ( x e. ran [,] |-> ( RR* \ x ) )

Assertion

Ref	Expression
lecldbas	|- (ordTop ` <_ ) = ( topGen ` ( fi ` ran F ) )

Proof of Theorem lecldbas

Dummy variables a b c y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2453	. . . 4 |- ran ( y e. RR* |-> ( y (,] +oo ) ) = ran ( y e. RR* |-> ( y (,] +oo ) )
2		eqid 2453	. . . 4 |- ran ( y e. RR* |-> ( -oo [,) y ) ) = ran ( y e. RR* |-> ( -oo [,) y ) )
3	1, 2	leordtval2 20240	. . 3 |- (ordTop ` <_ ) = ( topGen ` ( fi ` ( ran ( y e. RR* |-> ( y (,] +oo ) ) u. ran ( y e. RR* |-> ( -oo [,) y ) ) ) ) )
4		fvex 5880	. . . 4 |- ( fi ` ran F ) e. _V
5		fvex 5880	. . . . . 6 |- (ordTop ` <_ ) e. _V
6		lecldbas.1	. . . . . . . 8 |- F = ( x e. ran [,] |-> ( RR* \ x ) )
7		iccf 11740	. . . . . . . . . . 11 |- [,] : ( RR* X. RR* ) --> ~P RR*
8		ffn 5733	. . . . . . . . . . 11 |- ( [,] : ( RR* X. RR* ) --> ~P RR* -> [,] Fn ( RR* X. RR* ) )
9	7, 8	ax-mp 5	. . . . . . . . . 10 |- [,] Fn ( RR* X. RR* )
10		ovelrn 6450	. . . . . . . . . 10 |- ( [,] Fn ( RR* X. RR* ) -> ( x e. ran [,] <-> E. a e. RR* E. b e. RR* x = ( a [,] b ) ) )
11	9, 10	ax-mp 5	. . . . . . . . 9 |- ( x e. ran [,] <-> E. a e. RR* E. b e. RR* x = ( a [,] b ) )
12		difeq2 3547	. . . . . . . . . . . 12 |- ( x = ( a [,] b ) -> ( RR* \ x ) = ( RR* \ ( a [,] b ) ) )
13		iccordt 20242	. . . . . . . . . . . . 13 |- ( a [,] b ) e. ( Clsd ` (ordTop ` <_ ) )
14		letopuni 20235	. . . . . . . . . . . . . 14 |- RR* = U. (ordTop ` <_ )
15	14	cldopn 20058	. . . . . . . . . . . . 13 |- ( ( a [,] b ) e. ( Clsd ` (ordTop ` <_ ) ) -> ( RR* \ ( a [,] b ) ) e. (ordTop ` <_ ) )
16	13, 15	ax-mp 5	. . . . . . . . . . . 12 |- ( RR* \ ( a [,] b ) ) e. (ordTop ` <_ )
17	12, 16	syl6eqel 2539	. . . . . . . . . . 11 |- ( x = ( a [,] b ) -> ( RR* \ x ) e. (ordTop ` <_ ) )
18	17	rexlimivw 2878	. . . . . . . . . 10 |- ( E. b e. RR* x = ( a [,] b ) -> ( RR* \ x ) e. (ordTop ` <_ ) )
19	18	rexlimivw 2878	. . . . . . . . 9 |- ( E. a e. RR* E. b e. RR* x = ( a [,] b ) -> ( RR* \ x ) e. (ordTop ` <_ ) )
20	11, 19	sylbi 199	. . . . . . . 8 |- ( x e. ran [,] -> ( RR* \ x ) e. (ordTop ` <_ ) )
21	6, 20	fmpti 6050	. . . . . . 7 |- F : ran [,] --> (ordTop ` <_ )
22		frn 5740	. . . . . . 7 |- ( F : ran [,] --> (ordTop ` <_ ) -> ran F C_ (ordTop ` <_ ) )
23	21, 22	ax-mp 5	. . . . . 6 |- ran F C_ (ordTop ` <_ )
24	5, 23	ssexi 4551	. . . . 5 |- ran F e. _V
25		eqid 2453	. . . . . . . 8 |- ( y e. RR* |-> ( y (,] +oo ) ) = ( y e. RR* |-> ( y (,] +oo ) )
26		mnfxr 11421	. . . . . . . . . . 11 |- -oo e. RR*
27		fnovrn 6449	. . . . . . . . . . 11 |- ( ( [,] Fn ( RR* X. RR* ) /\ -oo e. RR* /\ y e. RR* ) -> ( -oo [,] y ) e. ran [,] )
28	9, 26, 27	mp3an12 1356	. . . . . . . . . 10 |- ( y e. RR* -> ( -oo [,] y ) e. ran [,] )
29	26	a1i 11	. . . . . . . . . . . . . 14 |- ( y e. RR* -> -oo e. RR* )
30		id 22	. . . . . . . . . . . . . 14 |- ( y e. RR* -> y e. RR* )
31		pnfxr 11419	. . . . . . . . . . . . . . 15 |- +oo e. RR*
32	31	a1i 11	. . . . . . . . . . . . . 14 |- ( y e. RR* -> +oo e. RR* )
33		mnfle 11442	. . . . . . . . . . . . . 14 |- ( y e. RR* -> -oo <_ y )
34		pnfge 11439	. . . . . . . . . . . . . 14 |- ( y e. RR* -> y <_ +oo )
35		df-icc 11649	. . . . . . . . . . . . . . 15 |- [,] = ( a e. RR* , b e. RR* |-> { c e. RR* | ( a <_ c /\ c <_ b ) } )
36		df-ioc 11647	. . . . . . . . . . . . . . 15 |- (,] = ( a e. RR* , b e. RR* |-> { c e. RR* | ( a < c /\ c <_ b ) } )
37		xrltnle 9706	. . . . . . . . . . . . . . 15 |- ( ( y e. RR* /\ z e. RR* ) -> ( y < z <-> -. z <_ y ) )
38		xrletr 11462	. . . . . . . . . . . . . . 15 |- ( ( z e. RR* /\ y e. RR* /\ +oo e. RR* ) -> ( ( z <_ y /\ y <_ +oo ) -> z <_ +oo ) )
39		xrlelttr 11460	. . . . . . . . . . . . . . . 16 |- ( ( -oo e. RR* /\ y e. RR* /\ z e. RR* ) -> ( ( -oo <_ y /\ y < z ) -> -oo < z ) )
40		xrltle 11455	. . . . . . . . . . . . . . . . 17 |- ( ( -oo e. RR* /\ z e. RR* ) -> ( -oo < z -> -oo <_ z ) )
41	40	3adant2 1028	. . . . . . . . . . . . . . . 16 |- ( ( -oo e. RR* /\ y e. RR* /\ z e. RR* ) -> ( -oo < z -> -oo <_ z ) )
42	39, 41	syld 45	. . . . . . . . . . . . . . 15 |- ( ( -oo e. RR* /\ y e. RR* /\ z e. RR* ) -> ( ( -oo <_ y /\ y < z ) -> -oo <_ z ) )
43	35, 36, 37, 35, 38, 42	ixxun 11658	. . . . . . . . . . . . . 14 |- ( ( ( -oo e. RR* /\ y e. RR* /\ +oo e. RR* ) /\ ( -oo <_ y /\ y <_ +oo ) ) -> ( ( -oo [,] y ) u. ( y (,] +oo ) ) = ( -oo [,] +oo ) )
44	29, 30, 32, 33, 34, 43	syl32anc 1277	. . . . . . . . . . . . 13 |- ( y e. RR* -> ( ( -oo [,] y ) u. ( y (,] +oo ) ) = ( -oo [,] +oo ) )
45		iccmax 11717	. . . . . . . . . . . . 13 |- ( -oo [,] +oo ) = RR*
46	44, 45	syl6eq 2503	. . . . . . . . . . . 12 |- ( y e. RR* -> ( ( -oo [,] y ) u. ( y (,] +oo ) ) = RR* )
47		iccssxr 11724	. . . . . . . . . . . . 13 |- ( -oo [,] y ) C_ RR*
48	35, 36, 37	ixxdisj 11657	. . . . . . . . . . . . . 14 |- ( ( -oo e. RR* /\ y e. RR* /\ +oo e. RR* ) -> ( ( -oo [,] y ) i^i ( y (,] +oo ) ) = (/) )
49	26, 31, 48	mp3an13 1357	. . . . . . . . . . . . 13 |- ( y e. RR* -> ( ( -oo [,] y ) i^i ( y (,] +oo ) ) = (/) )
50		uneqdifeq 3858	. . . . . . . . . . . . 13 |- ( ( ( -oo [,] y ) C_ RR* /\ ( ( -oo [,] y ) i^i ( y (,] +oo ) ) = (/) ) -> ( ( ( -oo [,] y ) u. ( y (,] +oo ) ) = RR* <-> ( RR* \ ( -oo [,] y ) ) = ( y (,] +oo ) ) )
51	47, 49, 50	sylancr 670	. . . . . . . . . . . 12 |- ( y e. RR* -> ( ( ( -oo [,] y ) u. ( y (,] +oo ) ) = RR* <-> ( RR* \ ( -oo [,] y ) ) = ( y (,] +oo ) ) )
52	46, 51	mpbid 214	. . . . . . . . . . 11 |- ( y e. RR* -> ( RR* \ ( -oo [,] y ) ) = ( y (,] +oo ) )
53	52	eqcomd 2459	. . . . . . . . . 10 |- ( y e. RR* -> ( y (,] +oo ) = ( RR* \ ( -oo [,] y ) ) )
54		difeq2 3547	. . . . . . . . . . . 12 |- ( x = ( -oo [,] y ) -> ( RR* \ x ) = ( RR* \ ( -oo [,] y ) ) )
55	54	eqeq2d 2463	. . . . . . . . . . 11 |- ( x = ( -oo [,] y ) -> ( ( y (,] +oo ) = ( RR* \ x ) <-> ( y (,] +oo ) = ( RR* \ ( -oo [,] y ) ) ) )
56	55	rspcev 3152	. . . . . . . . . 10 |- ( ( ( -oo [,] y ) e. ran [,] /\ ( y (,] +oo ) = ( RR* \ ( -oo [,] y ) ) ) -> E. x e. ran [,] ( y (,] +oo ) = ( RR* \ x ) )
57	28, 53, 56	syl2anc 667	. . . . . . . . 9 |- ( y e. RR* -> E. x e. ran [,] ( y (,] +oo ) = ( RR* \ x ) )
58		xrex 11306	. . . . . . . . . . 11 |- RR* e. _V
59		difexg 4554	. . . . . . . . . . 11 |- ( RR* e. _V -> ( RR* \ x ) e. _V )
60	58, 59	ax-mp 5	. . . . . . . . . 10 |- ( RR* \ x ) e. _V
61	6, 60	elrnmpti 5088	. . . . . . . . 9 |- ( ( y (,] +oo ) e. ran F <-> E. x e. ran [,] ( y (,] +oo ) = ( RR* \ x ) )
62	57, 61	sylibr 216	. . . . . . . 8 |- ( y e. RR* -> ( y (,] +oo ) e. ran F )
63	25, 62	fmpti 6050	. . . . . . 7 |- ( y e. RR* |-> ( y (,] +oo ) ) : RR* --> ran F
64		frn 5740	. . . . . . 7 |- ( ( y e. RR* |-> ( y (,] +oo ) ) : RR* --> ran F -> ran ( y e. RR* |-> ( y (,] +oo ) ) C_ ran F )
65	63, 64	ax-mp 5	. . . . . 6 |- ran ( y e. RR* |-> ( y (,] +oo ) ) C_ ran F
66		eqid 2453	. . . . . . . 8 |- ( y e. RR* |-> ( -oo [,) y ) ) = ( y e. RR* |-> ( -oo [,) y ) )
67		fnovrn 6449	. . . . . . . . . . 11 |- ( ( [,] Fn ( RR* X. RR* ) /\ y e. RR* /\ +oo e. RR* ) -> ( y [,] +oo ) e. ran [,] )
68	9, 31, 67	mp3an13 1357	. . . . . . . . . 10 |- ( y e. RR* -> ( y [,] +oo ) e. ran [,] )
69		df-ico 11648	. . . . . . . . . . . . . . 15 |- [,) = ( a e. RR* , b e. RR* |-> { c e. RR* | ( a <_ c /\ c < b ) } )
70		xrlenlt 9704	. . . . . . . . . . . . . . 15 |- ( ( y e. RR* /\ z e. RR* ) -> ( y <_ z <-> -. z < y ) )
71		xrltletr 11461	. . . . . . . . . . . . . . . 16 |- ( ( z e. RR* /\ y e. RR* /\ +oo e. RR* ) -> ( ( z < y /\ y <_ +oo ) -> z < +oo ) )
72		xrltle 11455	. . . . . . . . . . . . . . . . 17 |- ( ( z e. RR* /\ +oo e. RR* ) -> ( z < +oo -> z <_ +oo ) )
73	72	3adant2 1028	. . . . . . . . . . . . . . . 16 |- ( ( z e. RR* /\ y e. RR* /\ +oo e. RR* ) -> ( z < +oo -> z <_ +oo ) )
74	71, 73	syld 45	. . . . . . . . . . . . . . 15 |- ( ( z e. RR* /\ y e. RR* /\ +oo e. RR* ) -> ( ( z < y /\ y <_ +oo ) -> z <_ +oo ) )
75		xrletr 11462	. . . . . . . . . . . . . . 15 |- ( ( -oo e. RR* /\ y e. RR* /\ z e. RR* ) -> ( ( -oo <_ y /\ y <_ z ) -> -oo <_ z ) )
76	69, 35, 70, 35, 74, 75	ixxun 11658	. . . . . . . . . . . . . 14 |- ( ( ( -oo e. RR* /\ y e. RR* /\ +oo e. RR* ) /\ ( -oo <_ y /\ y <_ +oo ) ) -> ( ( -oo [,) y ) u. ( y [,] +oo ) ) = ( -oo [,] +oo ) )
77	29, 30, 32, 33, 34, 76	syl32anc 1277	. . . . . . . . . . . . 13 |- ( y e. RR* -> ( ( -oo [,) y ) u. ( y [,] +oo ) ) = ( -oo [,] +oo ) )
78		uncom 3580	. . . . . . . . . . . . 13 |- ( ( -oo [,) y ) u. ( y [,] +oo ) ) = ( ( y [,] +oo ) u. ( -oo [,) y ) )
79	77, 78, 45	3eqtr3g 2510	. . . . . . . . . . . 12 |- ( y e. RR* -> ( ( y [,] +oo ) u. ( -oo [,) y ) ) = RR* )
80		iccssxr 11724	. . . . . . . . . . . . 13 |- ( y [,] +oo ) C_ RR*
81		incom 3627	. . . . . . . . . . . . . 14 |- ( ( y [,] +oo ) i^i ( -oo [,) y ) ) = ( ( -oo [,) y ) i^i ( y [,] +oo ) )
82	69, 35, 70	ixxdisj 11657	. . . . . . . . . . . . . . 15 |- ( ( -oo e. RR* /\ y e. RR* /\ +oo e. RR* ) -> ( ( -oo [,) y ) i^i ( y [,] +oo ) ) = (/) )
83	26, 31, 82	mp3an13 1357	. . . . . . . . . . . . . 14 |- ( y e. RR* -> ( ( -oo [,) y ) i^i ( y [,] +oo ) ) = (/) )
84	81, 83	syl5eq 2499	. . . . . . . . . . . . 13 |- ( y e. RR* -> ( ( y [,] +oo ) i^i ( -oo [,) y ) ) = (/) )
85		uneqdifeq 3858	. . . . . . . . . . . . 13 |- ( ( ( y [,] +oo ) C_ RR* /\ ( ( y [,] +oo ) i^i ( -oo [,) y ) ) = (/) ) -> ( ( ( y [,] +oo ) u. ( -oo [,) y ) ) = RR* <-> ( RR* \ ( y [,] +oo ) ) = ( -oo [,) y ) ) )
86	80, 84, 85	sylancr 670	. . . . . . . . . . . 12 |- ( y e. RR* -> ( ( ( y [,] +oo ) u. ( -oo [,) y ) ) = RR* <-> ( RR* \ ( y [,] +oo ) ) = ( -oo [,) y ) ) )
87	79, 86	mpbid 214	. . . . . . . . . . 11 |- ( y e. RR* -> ( RR* \ ( y [,] +oo ) ) = ( -oo [,) y ) )
88	87	eqcomd 2459	. . . . . . . . . 10 |- ( y e. RR* -> ( -oo [,) y ) = ( RR* \ ( y [,] +oo ) ) )
89		difeq2 3547	. . . . . . . . . . . 12 |- ( x = ( y [,] +oo ) -> ( RR* \ x ) = ( RR* \ ( y [,] +oo ) ) )
90	89	eqeq2d 2463	. . . . . . . . . . 11 |- ( x = ( y [,] +oo ) -> ( ( -oo [,) y ) = ( RR* \ x ) <-> ( -oo [,) y ) = ( RR* \ ( y [,] +oo ) ) ) )
91	90	rspcev 3152	. . . . . . . . . 10 |- ( ( ( y [,] +oo ) e. ran [,] /\ ( -oo [,) y ) = ( RR* \ ( y [,] +oo ) ) ) -> E. x e. ran [,] ( -oo [,) y ) = ( RR* \ x ) )
92	68, 88, 91	syl2anc 667	. . . . . . . . 9 |- ( y e. RR* -> E. x e. ran [,] ( -oo [,) y ) = ( RR* \ x ) )
93	6, 60	elrnmpti 5088	. . . . . . . . 9 |- ( ( -oo [,) y ) e. ran F <-> E. x e. ran [,] ( -oo [,) y ) = ( RR* \ x ) )
94	92, 93	sylibr 216	. . . . . . . 8 |- ( y e. RR* -> ( -oo [,) y ) e. ran F )
95	66, 94	fmpti 6050	. . . . . . 7 |- ( y e. RR* |-> ( -oo [,) y ) ) : RR* --> ran F
96		frn 5740	. . . . . . 7 |- ( ( y e. RR* |-> ( -oo [,) y ) ) : RR* --> ran F -> ran ( y e. RR* |-> ( -oo [,) y ) ) C_ ran F )
97	95, 96	ax-mp 5	. . . . . 6 |- ran ( y e. RR* |-> ( -oo [,) y ) ) C_ ran F
98	65, 97	unssi 3611	. . . . 5 |- ( ran ( y e. RR* |-> ( y (,] +oo ) ) u. ran ( y e. RR* |-> ( -oo [,) y ) ) ) C_ ran F
99		fiss 7943	. . . . 5 |- ( ( ran F e. _V /\ ( ran ( y e. RR* |-> ( y (,] +oo ) ) u. ran ( y e. RR* |-> ( -oo [,) y ) ) ) C_ ran F ) -> ( fi ` ( ran ( y e. RR* |-> ( y (,] +oo ) ) u. ran ( y e. RR* |-> ( -oo [,) y ) ) ) ) C_ ( fi ` ran F ) )
100	24, 98, 99	mp2an 679	. . . 4 |- ( fi ` ( ran ( y e. RR* |-> ( y (,] +oo ) ) u. ran ( y e. RR* |-> ( -oo [,) y ) ) ) ) C_ ( fi ` ran F )
101		tgss 19996	. . . 4 |- ( ( ( fi ` ran F ) e. _V /\ ( fi ` ( ran ( y e. RR* |-> ( y (,] +oo ) ) u. ran ( y e. RR* |-> ( -oo [,) y ) ) ) ) C_ ( fi ` ran F ) ) -> ( topGen ` ( fi ` ( ran ( y e. RR* |-> ( y (,] +oo ) ) u. ran ( y e. RR* |-> ( -oo [,) y ) ) ) ) ) C_ ( topGen ` ( fi ` ran F ) ) )
102	4, 100, 101	mp2an 679	. . 3 |- ( topGen ` ( fi ` ( ran ( y e. RR* |-> ( y (,] +oo ) ) u. ran ( y e. RR* |-> ( -oo [,) y ) ) ) ) ) C_ ( topGen ` ( fi ` ran F ) )
103	3, 102	eqsstri 3464	. 2 |- (ordTop ` <_ ) C_ ( topGen ` ( fi ` ran F ) )
104		letop 20234	. . 3 |- (ordTop ` <_ ) e. Top
105		tgfiss 20019	. . 3 |- ( ( (ordTop ` <_ ) e. Top /\ ran F C_ (ordTop ` <_ ) ) -> ( topGen ` ( fi ` ran F ) ) C_ (ordTop ` <_ ) )
106	104, 23, 105	mp2an 679	. 2 |- ( topGen ` ( fi ` ran F ) ) C_ (ordTop ` <_ )
107	103, 106	eqssi 3450	1 |- (ordTop ` <_ ) = ( topGen ` ( fi ` ran F ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 986   = wceq 1446   e. wcel 1889   E.wrex 2740   _Vcvv 3047   \ cdif 3403   u. cun 3404   i^i cin 3405   C_ wss 3406   (/)c0 3733   ~Pcpw 3953   class class class wbr 4405   |-> cmpt 4464   X. cxp 4835   ran crn 4838   Fn wfn 5580   -->wf 5581   ` cfv 5585  (class class class)co 6295   ficfi 7929   +oocpnf 9677   -oocmnf 9678   RR*cxr 9679   < clt 9680   <_ cle 9681   (,]cioc 11643   [,)cico 11644   [,]cicc 11645   topGenctg 15348  ordTopcordt 15409   Topctop 19929   Clsdccld 20043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-iin 4284  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fi 7930  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-ioc 11647  df-ico 11648  df-icc 11649  df-topgen 15354  df-ordt 15411  df-ps 16458  df-tsr 16459  df-top 19933  df-bases 19934  df-topon 19935  df-cld 20046

This theorem is referenced by: (None)