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Theorem islp2 16877
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 16872 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( limPt `  J ) `  S )  <->  P  e.  ( ( cls `  J
) `  ( S  \  { P } ) ) ) )
323adant3 975 . 2  |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  X )  ->  ( P  e.  ( ( limPt `  J ) `  S )  <->  P  e.  ( ( cls `  J
) `  ( S  \  { P } ) ) ) )
4 ssdifss 3307 . . 3  |-  ( S 
C_  X  ->  ( S  \  { P }
)  C_  X )
51neindisj2 16860 . . 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 1216 . 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 244 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    \ cdif 3149    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   U.cuni 3827   ` cfv 5255   Topctop 16631   clsccl 16755   neicnei 16834   limPtclp 16866
This theorem is referenced by:  clslp  16879  lpbl  18049  reperflem  18323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-top 16636  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868
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