Theorem sncld 7787

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 7705	. . . . 5 |- ((J e. Top /\ {P} (_ X) -> (x e. ((cls` J)` {P}) <-> (x e. X /\ A.y e. J (x e. y -> (y i^i {P}) =/= (/)))))
3		haustop 7786	. . . . 5 |- (J e. Haus -> J e. Top)
4		snssi 2466	. . . . 5 |- (P e. X -> {P} (_ X)
5	2, 3, 4	syl2an 454	. . . 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 7784	. . . . . . . . . . . . . 14 |- ((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 2304	. . . . . . . . . . . . . . . . . . . 20 |- (((y i^i {P}) (_ (y i^i n) /\ (y i^i n) = (/)) -> (y i^i {P}) = (/))
8		snssi 2466	. . . . . . . . . . . . . . . . . . . . 21 |- (P e. n -> {P} (_ n)
9		sslin 2235	. . . . . . . . . . . . . . . . . . . . 21 |- ({P} (_ n -> (y i^i {P}) (_ (y i^i n))
10	8, 9	syl 10	. . . . . . . . . . . . . . . . . . . 20 |- (P e. n -> (y i^i {P}) (_ (y i^i n))
11	7, 10	sylan 448	. . . . . . . . . . . . . . . . . . 19 |- ((P e. n /\ (y i^i n) = (/)) -> (y i^i {P}) = (/))
12	11	anim2i 335	. . . . . . . . . . . . . . . . . 18 |- ((x e. y /\ (P e. n /\ (y i^i n) = (/))) -> (x e. y /\ (y i^i {P}) = (/)))
13	12	3impb 829	. . . . . . . . . . . . . . . . 17 |- ((x e. y /\ P e. n /\ (y i^i n) = (/)) -> (x e. y /\ (y i^i {P}) = (/)))
14	13	a1i 8	. . . . . . . . . . . . . . . 16 |- (n e. J -> ((x e. y /\ P e. n /\ (y i^i n) = (/)) -> (x e. y /\ (y i^i {P}) = (/))))
15	14	r19.23aiv 1743	. . . . . . . . . . . . . . 15 |- (E.n e. J (x e. y /\ P e. n /\ (y i^i n) = (/)) -> (x e. y /\ (y i^i {P}) = (/)))
16	15	r19.22si 1734	. . . . . . . . . . . . . 14 |- (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 10	. . . . . . . . . . . . 13 |- ((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 851	. . . . . . . . . . . 12 |- (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 362	. . . . . . . . . . 11 |- (((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 489	. . . . . . . . . 10 |- (((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 1632	. . . . . . . . 9 |- (((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 1587	. . . . . . . . . . . 12 |- ((y i^i {P}) =/= (/) <-> -. (y i^i {P}) = (/))
23	22	imbi2i 185	. . . . . . . . . . 11 |- ((x e. y -> (y i^i {P}) =/= (/)) <-> (x e. y -> -. (y i^i {P}) = (/)))
24	23	ralbii 1667	. . . . . . . . . 10 |- (A.y e. J (x e. y -> (y i^i {P}) =/= (/)) <-> A.y e. J (x e. y -> -. (y i^i {P}) = (/)))
25		ralinexa 1683	. . . . . . . . . 10 |- (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	bitr 173	. . . . . . . . 9 |- (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 206	. . . . . . . 8 |- (((J e. Haus /\ P e. X) /\ x e. X) -> (A.y e. J (x e. y -> (y i^i {P}) =/= (/)) -> x = P))
28	27	ex 373	. . . . . . 7 |- ((J e. Haus /\ P e. X) -> (x e. X -> (A.y e. J (x e. y -> (y i^i {P}) =/= (/)) -> x = P)))
29	28	imp3a 361	. . . . . 6 |- ((J e. Haus /\ P e. X) -> ((x e. X /\ A.y e. J (x e. y -> (y i^i {P}) =/= (/))) -> x = P))
30		eleq1a 1543	. . . . . . . 8 |- (P e. X -> (x = P -> x e. X))
31	30	adantl 388	. . . . . . 7 |- ((J e. Haus /\ P e. X) -> (x = P -> x e. X))
32		eleq1 1534	. . . . . . . . . . . 12 |- (x = P -> (x e. y <-> P e. y))
33	32	biimpd 153	. . . . . . . . . . 11 |- (x = P -> (x e. y -> P e. y))
34		disjsn 2441	. . . . . . . . . . . 12 |- ((y i^i {P}) = (/) <-> -. P e. y)
35	34	necon2abii 1620	. . . . . . . . . . 11 |- (P e. y <-> (y i^i {P}) =/= (/))
36	33, 35	syl6ib 212	. . . . . . . . . 10 |- (x = P -> (x e. y -> (y i^i {P}) =/= (/)))
37	36	adantr 389	. . . . . . . . 9 |- ((x = P /\ y e. J) -> (x e. y -> (y i^i {P}) =/= (/)))
38	37	r19.21aiva 1714	. . . . . . . 8 |- (x = P -> A.y e. J (x e. y -> (y i^i {P}) =/= (/)))
39	38	a1i 8	. . . . . . 7 |- ((J e. Haus /\ P e. X) -> (x = P -> A.y e. J (x e. y -> (y i^i {P}) =/= (/))))
40	31, 39	jcad 600	. . . . . 6 |- ((J e. Haus /\ P e. X) -> (x = P -> (x e. X /\ A.y e. J (x e. y -> (y i^i {P}) =/= (/)))))
41	29, 40	impbid 516	. . . . 5 |- ((J e. Haus /\ P e. X) -> ((x e. X /\ A.y e. J (x e. y -> (y i^i {P}) =/= (/))) <-> x = P))
42		elsn 2421	. . . . 5 |- (x e. {P} <-> x = P)
43	41, 42	syl6bbr 538	. . . 4 |- ((J e. Haus /\ P e. X) -> ((x e. X /\ A.y e. J (x e. y -> (y i^i {P}) =/= (/))) <-> x e. {P}))
44	5, 43	bitrd 528	. . 3 |- ((J e. Haus /\ P e. X) -> (x e. ((cls` J)` {P}) <-> x e. {P}))
45	44	eqrdv 1473	. 2 |- ((J e. Haus /\ P e. X) -> ((cls` J)` {P}) = {P})
46	1	iscld3 7695	. . 3 |- ((J e. Top /\ {P} (_ X) -> ({P} e. (Clsd` J) <-> ((cls` J)` {P}) = {P}))
47	46, 3, 4	syl2an 454	. 2 |- ((J e. Haus /\ P e. X) -> ({P} e. (Clsd` J) <-> ((cls` J)` {P}) = {P}))
48	45, 47	mpbird 196	1 |- ((J e. Haus /\ P e. X) -> {P} e. (Clsd` J))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958   =/= wne 1585  A.wral 1645  E.wrex 1646   i^i cin 2046   (_ wss 2047  (/)c0 2280  {csn 2409  U.cuni 2503  ` cfv 3182  Topctop 7588  Clsdccld 7660  clsccl 7662  Hauscha 7781

This theorem is referenced by:  dnsconst 7788  t2t1 10616

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-iin 2569  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-top 7592  df-cld 7663  df-ntr 7664  df-cls 7665  df-haus 7782

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