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Theorem neindisj 16870
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 16812 . . . . . . 7  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
32sseld 3192 . . . . . 6  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( cls `  J
) `  S )  ->  P  e.  X ) )
43impr 602 . . . . 5  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  P  e.  ( ( cls `  J ) `  S ) ) )  ->  P  e.  X
)
51isneip 16858 . . . . 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 456 . . . 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 938 . . . . . . . . . . . 12  |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
)  <->  ( J  e. 
Top  /\  ( S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
) ) )
81clsndisj 16828 . . . . . . . . . . . 12  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  ( ( cls `  J ) `  S ) )  /\  ( g  e.  J  /\  P  e.  g
) )  ->  (
g  i^i  S )  =/=  (/) )
97, 8sylanbr 459 . . . . . . . . . . 11  |-  ( ( ( J  e.  Top  /\  ( S  C_  X  /\  P  e.  (
( cls `  J
) `  S )
) )  /\  (
g  e.  J  /\  P  e.  g )
)  ->  ( g  i^i  S )  =/=  (/) )
109anassrs 629 . . . . . . . . . 10  |-  ( ( ( ( J  e. 
Top  /\  ( S  C_  X  /\  P  e.  ( ( cls `  J
) `  S )
) )  /\  g  e.  J )  /\  P  e.  g )  ->  (
g  i^i  S )  =/=  (/) )
1110adantllr 699 . . . . . . . . 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 697 . . . . . . . 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 3517 . . . . . . . . . . 11  |-  ( ( g  C_  N  /\  ( N  i^i  S )  =  (/) )  ->  (
g  i^i  S )  =  (/) )
1413ex 423 . . . . . . . . . 10  |-  ( g 
C_  N  ->  (
( N  i^i  S
)  =  (/)  ->  (
g  i^i  S )  =  (/) ) )
1514necon3d 2497 . . . . . . . . 9  |-  ( g 
C_  N  ->  (
( g  i^i  S
)  =/=  (/)  ->  ( N  i^i  S )  =/=  (/) ) )
1615ad2antll 709 . . . . . . . 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 14 . . . . . . 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 423 . . . . . 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 2680 . . . . 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 586 . . . 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 206 . . 3  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  P  e.  ( ( cls `  J ) `  S ) ) )  ->  ( N  e.  ( ( nei `  J
) `  { P } )  ->  ( N  i^i  S )  =/=  (/) ) )
2221exp32 588 . 2  |-  ( J  e.  Top  ->  ( S  C_  X  ->  ( P  e.  ( ( cls `  J ) `  S )  ->  ( N  e.  ( ( nei `  J ) `  { P } )  -> 
( N  i^i  S
)  =/=  (/) ) ) ) )
2322imp43 578 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557    i^i cin 3164    C_ wss 3165   (/)c0 3468   {csn 3653   U.cuni 3843   ` cfv 5271   Topctop 16647   clsccl 16771   neicnei 16850
This theorem is referenced by:  clslp  16895  flimclslem  17695  islimrs4  25685
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-top 16652  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851
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