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Theorem islp2 19939
Description: The predicate " P is a limit point of  S," in terms of neighborhoods. Definition of limit point in [Munkres] p. 97. Although Munkres uses open neighborhoods, it also works for our more general neighborhoods. (Contributed by NM, 26-Feb-2007.) (Proof shortened by Mario Carneiro, 25-Dec-2016.)
Hypothesis
Ref Expression
lpfval.1  |-  X  = 
U. J
Assertion
Ref Expression
islp2  |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  X )  ->  ( P  e.  ( ( limPt `  J ) `  S )  <->  A. n  e.  ( ( nei `  J
) `  { P } ) ( n  i^i  ( S  \  { P } ) )  =/=  (/) ) )
Distinct variable groups:    n, J    P, n    S, n    n, X

Proof of Theorem islp2
StepHypRef Expression
1 lpfval.1 . . . 4  |-  X  = 
U. J
21islp 19934 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( limPt `  J ) `  S )  <->  P  e.  ( ( cls `  J
) `  ( S  \  { P } ) ) ) )
323adant3 1017 . 2  |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  X )  ->  ( P  e.  ( ( limPt `  J ) `  S )  <->  P  e.  ( ( cls `  J
) `  ( S  \  { P } ) ) ) )
4 ssdifss 3574 . . 3  |-  ( S 
C_  X  ->  ( S  \  { P }
)  C_  X )
51neindisj2 19917 . . 3  |-  ( ( J  e.  Top  /\  ( S  \  { P } )  C_  X  /\  P  e.  X
)  ->  ( P  e.  ( ( cls `  J
) `  ( S  \  { P } ) )  <->  A. n  e.  ( ( nei `  J
) `  { P } ) ( n  i^i  ( S  \  { P } ) )  =/=  (/) ) )
64, 5syl3an2 1264 . 2  |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  X )  ->  ( P  e.  ( ( cls `  J ) `  ( S  \  { P } ) )  <->  A. n  e.  ( ( nei `  J
) `  { P } ) ( n  i^i  ( S  \  { P } ) )  =/=  (/) ) )
73, 6bitrd 253 1  |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  X )  ->  ( P  e.  ( ( limPt `  J ) `  S )  <->  A. n  e.  ( ( nei `  J
) `  { P } ) ( n  i^i  ( S  \  { P } ) )  =/=  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2754    \ cdif 3411    i^i cin 3413    C_ wss 3414   (/)c0 3738   {csn 3972   U.cuni 4191   ` cfv 5569   Topctop 19686   clsccl 19811   neicnei 19891   limPtclp 19928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-top 19691  df-cld 19812  df-ntr 19813  df-cls 19814  df-nei 19892  df-lp 19930
This theorem is referenced by:  clslp  19942  lpbl  21298  reperflem  21615  islptre  36993  islpcn  37013
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