Theorem neindisj 7672

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 (_ 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	. . . . . . . . 9 |- X = U.J
2	1	clsss3 7633	. . . . . . . 8 |- ((J e. Top /\ S (_ X) -> ((cls` J)` S) (_ X)
3	2	sseld 2057	. . . . . . 7 |- ((J e. Top /\ S (_ X) -> (P e. ((cls` J)` S) -> P e. X))
4	3	ex 373	. . . . . 6 |- (J e. Top -> (S (_ X -> (P e. ((cls` J)` S) -> P e. X)))
5	4	imp32 363	. . . . 5 |- ((J e. Top /\ (S (_ X /\ P e. ((cls` J)` S))) -> P e. X)
6	1	isneip 7661	. . . . 5 |- ((J e. Top /\ P e. X) -> (N e. ((nei` J)` {P}) <-> (N (_ X /\ E.g e. J (P e. g /\ g (_ N))))
7	5, 6	syldan 467	. . . 4 |- ((J e. Top /\ (S (_ X /\ P e. ((cls` J)` S))) -> (N e. ((nei` J)` {P}) <-> (N (_ X /\ E.g e. J (P e. g /\ g (_ N))))
8	1	clsndisj 7648	. . . . . . . . . . . . 13 |- (((J e. Top /\ S (_ X /\ P e. ((cls` J)` S)) /\ (g e. J /\ P e. g)) -> (g i^i S) =/= (/))
9		3anass 777	. . . . . . . . . . . . 13 |- ((J e. Top /\ S (_ X /\ P e. ((cls` J)` S)) <-> (J e. Top /\ (S (_ X /\ P e. ((cls` J)` S))))
10	8, 9	sylanbr 450	. . . . . . . . . . . 12 |- (((J e. Top /\ (S (_ X /\ P e. ((cls` J)` S))) /\ (g e. J /\ P e. g)) -> (g i^i S) =/= (/))
11	10	anassrs 441	. . . . . . . . . . 11 |- ((((J e. Top /\ (S (_ X /\ P e. ((cls` J)` S))) /\ g e. J) /\ P e. g) -> (g i^i S) =/= (/))
12	11	adantllr 397	. . . . . . . . . 10 |- (((((J e. Top /\ (S (_ X /\ P e. ((cls` J)` S))) /\ N (_ X) /\ g e. J) /\ P e. g) -> (g i^i S) =/= (/))
13	12	adantrr 395	. . . . . . . . 9 |- (((((J e. Top /\ (S (_ X /\ P e. ((cls` J)` S))) /\ N (_ X) /\ g e. J) /\ (P e. g /\ g (_ N)) -> (g i^i S) =/= (/))
14		ssdisj 2308	. . . . . . . . . . . 12 |- ((g (_ N /\ (N i^i S) = (/)) -> (g i^i S) = (/))
15	14	ex 373	. . . . . . . . . . 11 |- (g (_ N -> ((N i^i S) = (/) -> (g i^i S) = (/)))
16	15	necon3d 1596	. . . . . . . . . 10 |- (g (_ N -> ((g i^i S) =/= (/) -> (N i^i S) =/= (/)))
17	16	ad2antll 407	. . . . . . . . 9 |- (((((J e. Top /\ (S (_ X /\ P e. ((cls` J)` S))) /\ N (_ X) /\ g e. J) /\ (P e. g /\ g (_ N)) -> ((g i^i S) =/= (/) -> (N i^i S) =/= (/)))
18	13, 17	mpd 26	. . . . . . . 8 |- (((((J e. Top /\ (S (_ X /\ P e. ((cls` J)` S))) /\ N (_ X) /\ g e. J) /\ (P e. g /\ g (_ N)) -> (N i^i S) =/= (/))
19	18	ex 373	. . . . . . 7 |- ((((J e. Top /\ (S (_ X /\ P e. ((cls` J)` S))) /\ N (_ X) /\ g e. J) -> ((P e. g /\ g (_ N) -> (N i^i S) =/= (/)))
20	19	r19.23adva 1739	. . . . . 6 |- (((J e. Top /\ (S (_ X /\ P e. ((cls` J)` S))) /\ N (_ X) -> (E.g e. J (P e. g /\ g (_ N) -> (N i^i S) =/= (/)))
21	20	ex 373	. . . . 5 |- ((J e. Top /\ (S (_ X /\ P e. ((cls` J)` S))) -> (N (_ X -> (E.g e. J (P e. g /\ g (_ N) -> (N i^i S) =/= (/))))
22	21	imp3a 361	. . . 4 |- ((J e. Top /\ (S (_ X /\ P e. ((cls` J)` S))) -> ((N (_ X /\ E.g e. J (P e. g /\ g (_ N)) -> (N i^i S) =/= (/)))
23	7, 22	sylbid 203	. . 3 |- ((J e. Top /\ (S (_ X /\ P e. ((cls` J)` S))) -> (N e. ((nei` J)` {P}) -> (N i^i S) =/= (/)))
24	23	exp32 377	. 2 |- (J e. Top -> (S (_ X -> (P e. ((cls` J)` S) -> (N e. ((nei` J)` {P}) -> (N i^i S) =/= (/)))))
25	24	imp43 370	1 |- (((J e. Top /\ S (_ X) /\ (P e. ((cls` J)` S) /\ N e. ((nei` J)` {P}))) -> (N i^i S) =/= (/))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955   =/= wne 1577  E.wrex 1638   i^i cin 2036   (_ wss 2037  (/)c0 2270  {csn 2399  U.cuni 2493  ` cfv 3172  Topctop 7530  clsccl 7604  neicnei 7653

This theorem is referenced by:  clslp 7689

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-iin 2559  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188  df-top 7534  df-cld 7605  df-ntr 7606  df-cls 7607  df-nei 7654

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