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Theorem lpdifsn 17197
Description:  P is a limit point of  S iff it is a limit point of  S  \  { P }. (Contributed by Mario Carneiro, 25-Dec-2016.)
Hypothesis
Ref Expression
lpfval.1  |-  X  = 
U. J
Assertion
Ref Expression
lpdifsn  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( limPt `  J ) `  S )  <->  P  e.  ( ( limPt `  J
) `  ( S  \  { P } ) ) ) )

Proof of Theorem lpdifsn
StepHypRef Expression
1 lpfval.1 . . 3  |-  X  = 
U. J
21islp 17194 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( limPt `  J ) `  S )  <->  P  e.  ( ( cls `  J
) `  ( S  \  { P } ) ) ) )
3 ssdifss 3470 . . . 4  |-  ( S 
C_  X  ->  ( S  \  { P }
)  C_  X )
41islp 17194 . . . 4  |-  ( ( J  e.  Top  /\  ( S  \  { P } )  C_  X
)  ->  ( P  e.  ( ( limPt `  J
) `  ( S  \  { P } ) )  <->  P  e.  (
( cls `  J
) `  ( ( S  \  { P }
)  \  { P } ) ) ) )
53, 4sylan2 461 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( limPt `  J ) `  ( S  \  { P } ) )  <->  P  e.  ( ( cls `  J
) `  ( ( S  \  { P }
)  \  { P } ) ) ) )
6 difabs 3597 . . . . 5  |-  ( ( S  \  { P } )  \  { P } )  =  ( S  \  { P } )
76fveq2i 5723 . . . 4  |-  ( ( cls `  J ) `
 ( ( S 
\  { P }
)  \  { P } ) )  =  ( ( cls `  J
) `  ( S  \  { P } ) )
87eleq2i 2499 . . 3  |-  ( P  e.  ( ( cls `  J ) `  (
( S  \  { P } )  \  { P } ) )  <->  P  e.  ( ( cls `  J
) `  ( S  \  { P } ) ) )
95, 8syl6bb 253 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( limPt `  J ) `  ( S  \  { P } ) )  <->  P  e.  ( ( cls `  J
) `  ( S  \  { P } ) ) ) )
102, 9bitr4d 248 1  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( limPt `  J ) `  S )  <->  P  e.  ( ( limPt `  J
) `  ( S  \  { P } ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    \ cdif 3309    C_ wss 3312   {csn 3806   U.cuni 4007   ` cfv 5446   Topctop 16948   clsccl 17072   limPtclp 17188
This theorem is referenced by:  perfdvf  19780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-top 16953  df-cld 17073  df-cls 17075  df-lp 17190
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