Theorem ssnei2 9006

Description: Any subset of X containing a neigborhood of a set is a neighborhood of this set. Based on Bourbaki TG I.3 Vi. (Contributed by FL, 2-Oct-2006.)

Hypothesis

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

Assertion

Ref	Expression
ssnei2	|- (((J e. Top /\ N e. ((nei` J)` S)) /\ (N C_ M /\ M C_ X)) -> M e. ((nei` J)` S))

Proof of Theorem ssnei2

Step	Hyp	Ref	Expression
1		neips.1	. . . . 5 |- X = U.J
2	1	neiss2 8992	. . . 4 |- ((J e. Top /\ N e. ((nei` J)` S)) -> S C_ X)
3	1	isnei 8994	. . . 4 |- ((J e. Top /\ S C_ X) -> (M e. ((nei` J)` S) <-> (M C_ X /\ E.g e. J (S C_ g /\ g C_ M))))
4	2, 3	syldan 516	. . 3 |- ((J e. Top /\ N e. ((nei` J)` S)) -> (M e. ((nei` J)` S) <-> (M C_ X /\ E.g e. J (S C_ g /\ g C_ M))))
5	4	adantr 425	. 2 |- (((J e. Top /\ N e. ((nei` J)` S)) /\ (N C_ M /\ M C_ X)) -> (M e. ((nei` J)` S) <-> (M C_ X /\ E.g e. J (S C_ g /\ g C_ M))))
6		simprr 451	. 2 |- (((J e. Top /\ N e. ((nei` J)` S)) /\ (N C_ M /\ M C_ X)) -> M C_ X)
7		sstr2 2623	. . . . . . . 8 |- (g C_ N -> (N C_ M -> g C_ M))
8	7	com12 14	. . . . . . 7 |- (N C_ M -> (g C_ N -> g C_ M))
9	8	anim2d 620	. . . . . 6 |- (N C_ M -> ((S C_ g /\ g C_ N) -> (S C_ g /\ g C_ M)))
10	9	reximdv 2202	. . . . 5 |- (N C_ M -> (E.g e. J (S C_ g /\ g C_ N) -> E.g e. J (S C_ g /\ g C_ M)))
11	10	impcom 378	. . . 4 |- ((E.g e. J (S C_ g /\ g C_ N) /\ N C_ M) -> E.g e. J (S C_ g /\ g C_ M))
12		neii2 8998	. . . 4 |- ((J e. Top /\ N e. ((nei` J)` S)) -> E.g e. J (S C_ g /\ g C_ N))
13	11, 12	sylan 497	. . 3 |- (((J e. Top /\ N e. ((nei` J)` S)) /\ N C_ M) -> E.g e. J (S C_ g /\ g C_ M))
14	13	adantrr 431	. 2 |- (((J e. Top /\ N e. ((nei` J)` S)) /\ (N C_ M /\ M C_ X)) -> E.g e. J (S C_ g /\ g C_ M))
15	5, 6, 14	mpbir2and 802	1 |- (((J e. Top /\ N e. ((nei` J)` S)) /\ (N C_ M /\ M C_ X)) -> M e. ((nei` J)` S))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wrex 2106   C_ wss 2593  U.cuni 3177  ` cfv 3998  Topctop 8857  neicnei 8988

This theorem is referenced by:  neifil 10302  cnpfillim 15589

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-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-v 2294  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-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-nei 8989

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