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Theorem lpdifsn 16891
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 16888 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( limPt `  J ) `  S )  <->  P  e.  ( ( cls `  J
) `  ( S  \  { P } ) ) ) )
3 ssdifss 3320 . . . 4  |-  ( S 
C_  X  ->  ( S  \  { P }
)  C_  X )
41islp 16888 . . . 4  |-  ( ( J  e.  Top  /\  ( S  \  { P } )  C_  X
)  ->  ( P  e.  ( ( limPt `  J
) `  ( S  \  { P } ) )  <->  P  e.  (
( cls `  J
) `  ( ( S  \  { P }
)  \  { P } ) ) ) )
53, 4sylan2 460 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( limPt `  J ) `  ( S  \  { P } ) )  <->  P  e.  ( ( cls `  J
) `  ( ( S  \  { P }
)  \  { P } ) ) ) )
6 difabs 3445 . . . . 5  |-  ( ( S  \  { P } )  \  { P } )  =  ( S  \  { P } )
76fveq2i 5544 . . . 4  |-  ( ( cls `  J ) `
 ( ( S 
\  { P }
)  \  { P } ) )  =  ( ( cls `  J
) `  ( S  \  { P } ) )
87eleq2i 2360 . . 3  |-  ( P  e.  ( ( cls `  J ) `  (
( S  \  { P } )  \  { P } ) )  <->  P  e.  ( ( cls `  J
) `  ( S  \  { P } ) ) )
95, 8syl6bb 252 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( limPt `  J ) `  ( S  \  { P } ) )  <->  P  e.  ( ( cls `  J
) `  ( S  \  { P } ) ) ) )
102, 9bitr4d 247 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 176    /\ wa 358    = wceq 1632    e. wcel 1696    \ cdif 3162    C_ wss 3165   {csn 3653   U.cuni 3843   ` cfv 5271   Topctop 16647   clsccl 16771   limPtclp 16882
This theorem is referenced by:  perfdvf  19269
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-top 16652  df-cld 16772  df-cls 16774  df-lp 16884
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