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Theorem islpi 19411
Description: A point belonging to a set's closure but not the set itself is a limit point. (Contributed by NM, 8-Nov-2007.)
Hypothesis
Ref Expression
lpfval.1  |-  X  = 
U. J
Assertion
Ref Expression
islpi  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  ( P  e.  ( ( cls `  J
) `  S )  /\  -.  P  e.  S
) )  ->  P  e.  ( ( limPt `  J
) `  S )
)

Proof of Theorem islpi
StepHypRef Expression
1 lpfval.1 . . . . . 6  |-  X  = 
U. J
21clslp 19410 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  ( S  u.  ( ( limPt `  J
) `  S )
) )
32eleq2d 2532 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( cls `  J
) `  S )  <->  P  e.  ( S  u.  ( ( limPt `  J
) `  S )
) ) )
4 elun 3640 . . . . 5  |-  ( P  e.  ( S  u.  ( ( limPt `  J
) `  S )
)  <->  ( P  e.  S  \/  P  e.  ( ( limPt `  J
) `  S )
) )
5 df-or 370 . . . . 5  |-  ( ( P  e.  S  \/  P  e.  ( ( limPt `  J ) `  S ) )  <->  ( -.  P  e.  S  ->  P  e.  ( ( limPt `  J ) `  S
) ) )
64, 5bitri 249 . . . 4  |-  ( P  e.  ( S  u.  ( ( limPt `  J
) `  S )
)  <->  ( -.  P  e.  S  ->  P  e.  ( ( limPt `  J
) `  S )
) )
73, 6syl6bb 261 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( cls `  J
) `  S )  <->  ( -.  P  e.  S  ->  P  e.  ( (
limPt `  J ) `  S ) ) ) )
87biimpd 207 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( cls `  J
) `  S )  ->  ( -.  P  e.  S  ->  P  e.  ( ( limPt `  J
) `  S )
) ) )
98imp32 433 1  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  ( P  e.  ( ( cls `  J
) `  S )  /\  -.  P  e.  S
) )  ->  P  e.  ( ( limPt `  J
) `  S )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1374    e. wcel 1762    u. cun 3469    C_ wss 3471   U.cuni 4240   ` cfv 5581   Topctop 19156   clsccl 19280   limPtclp 19396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-iin 4323  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-top 19161  df-cld 19281  df-ntr 19282  df-cls 19283  df-nei 19360  df-lp 19398
This theorem is referenced by: (None)
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