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Theorem islp2 16933
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 16928 . . 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 3341 . . 3  |-  ( S 
C_  X  ->  ( S  \  { P }
)  C_  X )
51neindisj2 16916 . . 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 1633    e. wcel 1701    =/= wne 2479   A.wral 2577    \ cdif 3183    i^i cin 3185    C_ wss 3186   (/)c0 3489   {csn 3674   U.cuni 3864   ` cfv 5292   Topctop 16687   clsccl 16811   neicnei 16890   limPtclp 16922
This theorem is referenced by:  clslp  16935  lpbl  18101  reperflem  18375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-top 16692  df-cld 16812  df-ntr 16813  df-cls 16814  df-nei 16891  df-lp 16924
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