Theorem neindisj 7731

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 7691	. . . . . . . 8 |- ((J e. Top /\ S (_ X) -> ((cls` J)` S) (_ X)
3	2	sseld 2067	. . . . . . 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 7720	. . . . 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 7706	. . . . . . . . . . . . 13 |- (((J e. Top /\ S (_ X /\ P e. ((cls` J)` S)) /\ (g e. J /\ P e. g)) -> (g i^i S) =/= (/))
9		3anass 779	. . . . . . . . . . . . 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 2318	. . . . . . . . . . . 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 1604	. . . . . . . . . 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 1747	. . . . . 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 775   = wceq 956   e. wcel 958   =/= wne 1585  E.wrex 1646   i^i cin 2046   (_ wss 2047  (/)c0 2280  {csn 2409  U.cuni 2503  ` cfv 3182  Topctop 7588  clsccl 7662  neicnei 7712

This theorem is referenced by:  clslp 7748

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-nei 7713

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