Theorem neindisj 7681

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 7641	. . . . . . . 8 |- ((J e. Top /\ S (_ X) -> ((cls` J)` S) (_ X)
3	2	sseld 2063	. . . . . . 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 7670	. . . . 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 7656	. . . . . . . . . . . . 13 |- (((J e. Top /\ S (_ X /\ P e. ((cls` J)` S)) /\ (g e. J /\ P e. g)) -> (g i^i S) =/= (/))
9		3anass 778	. . . . . . . . . . . . 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 2314	. . . . . . . . . . . 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 1601	. . . . . . . . . 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 1744	. . . . . 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 774   = wceq 954   e. wcel 956   =/= wne 1582  E.wrex 1643   i^i cin 2042   (_ wss 2043  (/)c0 2276  {csn 2405  U.cuni 2498  ` cfv 3177  Topctop 7538  clsccl 7612  neicnei 7662

This theorem is referenced by:  clslp 7698

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-iin 2564  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193  df-top 7542  df-cld 7613  df-ntr 7614  df-cls 7615  df-nei 7663

Copyright terms: Public domain