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Theorem lpdifsn 17132
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 17129 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( limPt `  J ) `  S )  <->  P  e.  ( ( cls `  J
) `  ( S  \  { P } ) ) ) )
3 ssdifss 3423 . . . 4  |-  ( S 
C_  X  ->  ( S  \  { P }
)  C_  X )
41islp 17129 . . . 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 3550 . . . . 5  |-  ( ( S  \  { P } )  \  { P } )  =  ( S  \  { P } )
76fveq2i 5673 . . . 4  |-  ( ( cls `  J ) `
 ( ( S 
\  { P }
)  \  { P } ) )  =  ( ( cls `  J
) `  ( S  \  { P } ) )
87eleq2i 2453 . . 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 1649    e. wcel 1717    \ cdif 3262    C_ wss 3265   {csn 3759   U.cuni 3959   ` cfv 5396   Topctop 16883   clsccl 17007   limPtclp 17123
This theorem is referenced by:  perfdvf  19659
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-top 16888  df-cld 17008  df-cls 17010  df-lp 17125
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