Theorem sncld 9064

Description: A singleton is closed in a Hausdorff space.

Hypothesis

Ref	Expression
sncld.1	|- X = U.J

Assertion

Ref	Expression
sncld	|- ((J e. Haus /\ P e. X) -> {P} e. (Clsd` J))

Proof of Theorem sncld

Step	Hyp	Ref	Expression
1		sncld.1	. . . . . 6 |- X = U.J
2	1	elcls2 8981	. . . . 5 |- ((J e. Top /\ {P} C_ X) -> (x e. ((cls` J)` {P}) <-> (x e. X /\ A.y e. J (x e. y -> (y i^i {P}) =/= (/)))))
3		haustop 9063	. . . . 5 |- (J e. Haus -> J e. Top)
4		snssi 3129	. . . . 5 |- (P e. X -> {P} C_ X)
5	2, 3, 4	syl2an 503	. . . 4 |- ((J e. Haus /\ P e. X) -> (x e. ((cls` J)` {P}) <-> (x e. X /\ A.y e. J (x e. y -> (y i^i {P}) =/= (/)))))
6	1	hausnei 9061	. . . . . . . . . . . . 13 |- ((J e. Haus /\ (x e. X /\ P e. X /\ x =/= P)) -> E.y e. J E.n e. J (x e. y /\ P e. n /\ (y i^i n) = (/)))
7		sseq0 2903	. . . . . . . . . . . . . . . . . . 19 |- (((y i^i {P}) C_ (y i^i n) /\ (y i^i n) = (/)) -> (y i^i {P}) = (/))
8		snssi 3129	. . . . . . . . . . . . . . . . . . . 20 |- (P e. n -> {P} C_ n)
9		sslin 2819	. . . . . . . . . . . . . . . . . . . 20 |- ({P} C_ n -> (y i^i {P}) C_ (y i^i n))
10	8, 9	syl 12	. . . . . . . . . . . . . . . . . . 19 |- (P e. n -> (y i^i {P}) C_ (y i^i n))
11	7, 10	sylan 497	. . . . . . . . . . . . . . . . . 18 |- ((P e. n /\ (y i^i n) = (/)) -> (y i^i {P}) = (/))
12	11	anim2i 362	. . . . . . . . . . . . . . . . 17 |- ((x e. y /\ (P e. n /\ (y i^i n) = (/))) -> (x e. y /\ (y i^i {P}) = (/)))
13	12	3impb 1063	. . . . . . . . . . . . . . . 16 |- ((x e. y /\ P e. n /\ (y i^i n) = (/)) -> (x e. y /\ (y i^i {P}) = (/)))
14	13	a1i 8	. . . . . . . . . . . . . . 15 |- (n e. J -> ((x e. y /\ P e. n /\ (y i^i n) = (/)) -> (x e. y /\ (y i^i {P}) = (/))))
15	14	r19.23aiv 2211	. . . . . . . . . . . . . 14 |- (E.n e. J (x e. y /\ P e. n /\ (y i^i n) = (/)) -> (x e. y /\ (y i^i {P}) = (/)))
16	15	reximi 2198	. . . . . . . . . . . . 13 |- (E.y e. J E.n e. J (x e. y /\ P e. n /\ (y i^i n) = (/)) -> E.y e. J (x e. y /\ (y i^i {P}) = (/)))
17	6, 16	syl 12	. . . . . . . . . . . 12 |- ((J e. Haus /\ (x e. X /\ P e. X /\ x =/= P)) -> E.y e. J (x e. y /\ (y i^i {P}) = (/)))
18	17	3exp2 1086	. . . . . . . . . . 11 |- (J e. Haus -> (x e. X -> (P e. X -> (x =/= P -> E.y e. J (x e. y /\ (y i^i {P}) = (/))))))
19	18	imp31 389	. . . . . . . . . 10 |- (((J e. Haus /\ x e. X) /\ P e. X) -> (x =/= P -> E.y e. J (x e. y /\ (y i^i {P}) = (/))))
20	19	an1rs 547	. . . . . . . . 9 |- (((J e. Haus /\ P e. X) /\ x e. X) -> (x =/= P -> E.y e. J (x e. y /\ (y i^i {P}) = (/))))
21	20	necon1bd 2080	. . . . . . . 8 |- (((J e. Haus /\ P e. X) /\ x e. X) -> (-. E.y e. J (x e. y /\ (y i^i {P}) = (/)) -> x = P))
22		df-ne 2019	. . . . . . . . . . 11 |- ((y i^i {P}) =/= (/) <-> -. (y i^i {P}) = (/))
23	22	imbi2i 202	. . . . . . . . . 10 |- ((x e. y -> (y i^i {P}) =/= (/)) <-> (x e. y -> -. (y i^i {P}) = (/)))
24	23	ralbii 2127	. . . . . . . . 9 |- (A.y e. J (x e. y -> (y i^i {P}) =/= (/)) <-> A.y e. J (x e. y -> -. (y i^i {P}) = (/)))
25		ralinexa 2143	. . . . . . . . 9 |- (A.y e. J (x e. y -> -. (y i^i {P}) = (/)) <-> -. E.y e. J (x e. y /\ (y i^i {P}) = (/)))
26	24, 25	bitri 190	. . . . . . . 8 |- (A.y e. J (x e. y -> (y i^i {P}) =/= (/)) <-> -. E.y e. J (x e. y /\ (y i^i {P}) = (/)))
27	21, 26	syl5ib 223	. . . . . . 7 |- (((J e. Haus /\ P e. X) /\ x e. X) -> (A.y e. J (x e. y -> (y i^i {P}) =/= (/)) -> x = P))
28	27	expimpd 404	. . . . . 6 |- ((J e. Haus /\ P e. X) -> ((x e. X /\ A.y e. J (x e. y -> (y i^i {P}) =/= (/))) -> x = P))
29		eleq1a 1966	. . . . . . . 8 |- (P e. X -> (x = P -> x e. X))
30	29	adantl 424	. . . . . . 7 |- ((J e. Haus /\ P e. X) -> (x = P -> x e. X))
31		eleq1 1957	. . . . . . . . . . . 12 |- (x = P -> (x e. y <-> P e. y))
32	31	biimpd 170	. . . . . . . . . . 11 |- (x = P -> (x e. y -> P e. y))
33		disjsn 3089	. . . . . . . . . . . 12 |- ((y i^i {P}) = (/) <-> -. P e. y)
34	33	necon2abii 2063	. . . . . . . . . . 11 |- (P e. y <-> (y i^i {P}) =/= (/))
35	32, 34	syl6ib 229	. . . . . . . . . 10 |- (x = P -> (x e. y -> (y i^i {P}) =/= (/)))
36	35	adantr 425	. . . . . . . . 9 |- ((x = P /\ y e. J) -> (x e. y -> (y i^i {P}) =/= (/)))
37	36	r19.21aiva 2176	. . . . . . . 8 |- (x = P -> A.y e. J (x e. y -> (y i^i {P}) =/= (/)))
38	37	a1i 8	. . . . . . 7 |- ((J e. Haus /\ P e. X) -> (x = P -> A.y e. J (x e. y -> (y i^i {P}) =/= (/))))
39	30, 38	jcad 661	. . . . . 6 |- ((J e. Haus /\ P e. X) -> (x = P -> (x e. X /\ A.y e. J (x e. y -> (y i^i {P}) =/= (/)))))
40	28, 39	impbid 574	. . . . 5 |- ((J e. Haus /\ P e. X) -> ((x e. X /\ A.y e. J (x e. y -> (y i^i {P}) =/= (/))) <-> x = P))
41		elsn 3058	. . . . 5 |- (x e. {P} <-> x = P)
42	40, 41	syl6bbr 597	. . . 4 |- ((J e. Haus /\ P e. X) -> ((x e. X /\ A.y e. J (x e. y -> (y i^i {P}) =/= (/))) <-> x e. {P}))
43	5, 42	bitrd 587	. . 3 |- ((J e. Haus /\ P e. X) -> (x e. ((cls` J)` {P}) <-> x e. {P}))
44	43	eqrdv 1882	. 2 |- ((J e. Haus /\ P e. X) -> ((cls` J)` {P}) = {P})
45	1	iscld3 8971	. . 3 |- ((J e. Top /\ {P} C_ X) -> ({P} e. (Clsd` J) <-> ((cls` J)` {P}) = {P}))
46	45, 3, 4	syl2an 503	. 2 |- ((J e. Haus /\ P e. X) -> ({P} e. (Clsd` J) <-> ((cls` J)` {P}) = {P}))
47	44, 46	mpbird 213	1 |- ((J e. Haus /\ P e. X) -> {P} e. (Clsd` J))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106   i^i cin 2592   C_ wss 2593  (/)c0 2875  {csn 3044  U.cuni 3177  ` cfv 3998  Topctop 8857  Clsdccld 8936  clsccl 8938  Hauscha 9058

This theorem is referenced by:  dnsconst 9065  t2t1 14949

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-iin 3258  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-top 8861  df-cld 8939  df-ntr 8940  df-cls 8941  df-haus 9059

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