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Theorem neindisj 19384
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. (Contributed by NM, 26-Feb-2007.)
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
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 neips.1 . . . . . . . 8  |-  X  = 
U. J
21clsss3 19326 . . . . . . 7  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
32sseld 3503 . . . . . 6  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( cls `  J
) `  S )  ->  P  e.  X ) )
43impr 619 . . . . 5  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  P  e.  ( ( cls `  J ) `  S ) ) )  ->  P  e.  X
)
51isneip 19372 . . . . 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 ) ) ) )
64, 5syldan 470 . . . 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 3anass 977 . . . . . . . . . . . 12  |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
)  <->  ( J  e. 
Top  /\  ( S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
) ) )
81clsndisj 19342 . . . . . . . . . . . 12  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  ( ( cls `  J ) `  S ) )  /\  ( g  e.  J  /\  P  e.  g
) )  ->  (
g  i^i  S )  =/=  (/) )
97, 8sylanbr 473 . . . . . . . . . . 11  |-  ( ( ( J  e.  Top  /\  ( S  C_  X  /\  P  e.  (
( cls `  J
) `  S )
) )  /\  (
g  e.  J  /\  P  e.  g )
)  ->  ( g  i^i  S )  =/=  (/) )
109anassrs 648 . . . . . . . . . 10  |-  ( ( ( ( J  e. 
Top  /\  ( S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
) )  /\  g  e.  J )  /\  P  e.  g )  ->  (
g  i^i  S )  =/=  (/) )
1110adantllr 718 . . . . . . . . 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 )  =/=  (/) )
1211adantrr 716 . . . . . . . 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 3876 . . . . . . . . . . 11  |-  ( ( g  C_  N  /\  ( N  i^i  S )  =  (/) )  ->  (
g  i^i  S )  =  (/) )
1413ex 434 . . . . . . . . . 10  |-  ( g 
C_  N  ->  (
( N  i^i  S
)  =  (/)  ->  (
g  i^i  S )  =  (/) ) )
1514necon3d 2691 . . . . . . . . 9  |-  ( g 
C_  N  ->  (
( g  i^i  S
)  =/=  (/)  ->  ( N  i^i  S )  =/=  (/) ) )
1615ad2antll 728 . . . . . . . 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 )  =/=  (/) ) )
1712, 16mpd 15 . . . . . . 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 )  =/=  (/) )
1817ex 434 . . . . . 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 )  =/=  (/) ) )
1918rexlimdva 2955 . . . . 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 )  =/=  (/) ) )
2019expimpd 603 . . . 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 )  =/=  (/) ) )
216, 20sylbid 215 . . 3  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  P  e.  ( ( cls `  J ) `  S ) ) )  ->  ( N  e.  ( ( nei `  J
) `  { P } )  ->  ( N  i^i  S )  =/=  (/) ) )
2221exp32 605 . 2  |-  ( J  e.  Top  ->  ( S  C_  X  ->  ( P  e.  ( ( cls `  J ) `  S )  ->  ( N  e.  ( ( nei `  J ) `  { P } )  -> 
( N  i^i  S
)  =/=  (/) ) ) ) )
2322imp43 595 1  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  ( P  e.  ( ( cls `  J
) `  S )  /\  N  e.  (
( nei `  J
) `  { P } ) ) )  ->  ( N  i^i  S )  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815    i^i cin 3475    C_ wss 3476   (/)c0 3785   {csn 4027   U.cuni 4245   ` cfv 5586   Topctop 19161   clsccl 19285   neicnei 19364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-top 19166  df-cld 19286  df-ntr 19287  df-cls 19288  df-nei 19365
This theorem is referenced by:  clslp  19415  flimclslem  20220  utop3cls  20489
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