Theorem neindisj 9007

Description: Any neighborhood of an element in the closure of a subset intersects the subset. Part of proof of Theorem 6.6 of [Munkres] p. 97.

Hypothesis

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

Assertion

Ref	Expression
neindisj	|- (((J e. Top /\ S C_ X) /\ (P e. ((cls` J)` S) /\ N e. ((nei` J)` {P}))) -> (N i^i S) =/= (/))

Proof of Theorem neindisj

Step	Hyp	Ref	Expression
1		neips.1	. . . . . . . 8 |- X = U.J
2	1	clsss3 8967	. . . . . . 7 |- ((J e. Top /\ S C_ X) -> ((cls` J)` S) C_ X)
3	2	sseld 2619	. . . . . 6 |- ((J e. Top /\ S C_ X) -> (P e. ((cls` J)` S) -> P e. X))
4	3	impr 422	. . . . 5 |- ((J e. Top /\ (S C_ X /\ P e. ((cls` J)` S))) -> P e. X)
5	1	isneip 8996	. . . . 5 |- ((J e. Top /\ P e. X) -> (N e. ((nei` J)` {P}) <-> (N C_ X /\ E.g e. J (P e. g /\ g C_ N))))
6	4, 5	syldan 516	. . . 4 |- ((J e. Top /\ (S C_ X /\ P e. ((cls` J)` S))) -> (N e. ((nei` J)` {P}) <-> (N C_ X /\ E.g e. J (P e. g /\ g C_ N))))
7	1	clsndisj 8982	. . . . . . . . . . . 12 |- (((J e. Top /\ S C_ X /\ P e. ((cls` J)` S)) /\ (g e. J /\ P e. g)) -> (g i^i S) =/= (/))
8		3anass 862	. . . . . . . . . . . 12 |- ((J e. Top /\ S C_ X /\ P e. ((cls` J)` S)) <-> (J e. Top /\ (S C_ X /\ P e. ((cls` J)` S))))
9	7, 8	sylanbr 499	. . . . . . . . . . 11 |- (((J e. Top /\ (S C_ X /\ P e. ((cls` J)` S))) /\ (g e. J /\ P e. g)) -> (g i^i S) =/= (/))
10	9	anassrs 489	. . . . . . . . . 10 |- ((((J e. Top /\ (S C_ X /\ P e. ((cls` J)` S))) /\ g e. J) /\ P e. g) -> (g i^i S) =/= (/))
11	10	adantllr 433	. . . . . . . . 9 |- (((((J e. Top /\ (S C_ X /\ P e. ((cls` J)` S))) /\ N C_ X) /\ g e. J) /\ P e. g) -> (g i^i S) =/= (/))
12	11	adantrr 431	. . . . . . . 8 |- (((((J e. Top /\ (S C_ X /\ P e. ((cls` J)` S))) /\ N C_ X) /\ g e. J) /\ (P e. g /\ g C_ N)) -> (g i^i S) =/= (/))
13		ssdisj 2923	. . . . . . . . . . 11 |- ((g C_ N /\ (N i^i S) = (/)) -> (g i^i S) = (/))
14	13	ex 402	. . . . . . . . . 10 |- (g C_ N -> ((N i^i S) = (/) -> (g i^i S) = (/)))
15	14	necon3d 2041	. . . . . . . . 9 |- (g C_ N -> ((g i^i S) =/= (/) -> (N i^i S) =/= (/)))
16	15	ad2antll 443	. . . . . . . 8 |- (((((J e. Top /\ (S C_ X /\ P e. ((cls` J)` S))) /\ N C_ X) /\ g e. J) /\ (P e. g /\ g C_ N)) -> ((g i^i S) =/= (/) -> (N i^i S) =/= (/)))
17	12, 16	mpd 29	. . . . . . 7 |- (((((J e. Top /\ (S C_ X /\ P e. ((cls` J)` S))) /\ N C_ X) /\ g e. J) /\ (P e. g /\ g C_ N)) -> (N i^i S) =/= (/))
18	17	ex 402	. . . . . 6 |- ((((J e. Top /\ (S C_ X /\ P e. ((cls` J)` S))) /\ N C_ X) /\ g e. J) -> ((P e. g /\ g C_ N) -> (N i^i S) =/= (/)))
19	18	r19.23adva 2216	. . . . 5 |- (((J e. Top /\ (S C_ X /\ P e. ((cls` J)` S))) /\ N C_ X) -> (E.g e. J (P e. g /\ g C_ N) -> (N i^i S) =/= (/)))
20	19	expimpd 404	. . . 4 |- ((J e. Top /\ (S C_ X /\ P e. ((cls` J)` S))) -> ((N C_ X /\ E.g e. J (P e. g /\ g C_ N)) -> (N i^i S) =/= (/)))
21	6, 20	sylbid 220	. . 3 |- ((J e. Top /\ (S C_ X /\ P e. ((cls` J)` S))) -> (N e. ((nei` J)` {P}) -> (N i^i S) =/= (/)))
22	21	exp32 408	. 2 |- (J e. Top -> (S C_ X -> (P e. ((cls` J)` S) -> (N e. ((nei` J)` {P}) -> (N i^i S) =/= (/)))))
23	22	imp43 397	1 |- (((J e. Top /\ S C_ X) /\ (P e. ((cls` J)` S) /\ N e. ((nei` J)` {P}))) -> (N i^i S) =/= (/))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  E.wrex 2106   i^i cin 2592   C_ wss 2593  (/)c0 2875  {csn 3044  U.cuni 3177  ` cfv 3998  Topctop 8857  clsccl 8938  neicnei 8988

This theorem is referenced by:  clslp 9024  flimcls 15588

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-nei 8989

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