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Theorem islpi 17215
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 17214 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  ( S  u.  ( ( limPt `  J
) `  S )
) )
32eleq2d 2505 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( cls `  J
) `  S )  <->  P  e.  ( S  u.  ( ( limPt `  J
) `  S )
) ) )
4 elun 3490 . . . . 5  |-  ( P  e.  ( S  u.  ( ( limPt `  J
) `  S )
)  <->  ( P  e.  S  \/  P  e.  ( ( limPt `  J
) `  S )
) )
5 df-or 361 . . . . 5  |-  ( ( P  e.  S  \/  P  e.  ( ( limPt `  J ) `  S ) )  <->  ( -.  P  e.  S  ->  P  e.  ( ( limPt `  J ) `  S
) ) )
64, 5bitri 242 . . . 4  |-  ( P  e.  ( S  u.  ( ( limPt `  J
) `  S )
)  <->  ( -.  P  e.  S  ->  P  e.  ( ( limPt `  J
) `  S )
) )
73, 6syl6bb 254 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( cls `  J
) `  S )  <->  ( -.  P  e.  S  ->  P  e.  ( (
limPt `  J ) `  S ) ) ) )
87biimpd 200 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( cls `  J
) `  S )  ->  ( -.  P  e.  S  ->  P  e.  ( ( limPt `  J
) `  S )
) ) )
98imp32 424 1  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  ( P  e.  ( ( cls `  J
) `  S )  /\  -.  P  e.  S
) )  ->  P  e.  ( ( limPt `  J
) `  S )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726    u. cun 3320    C_ wss 3322   U.cuni 4017   ` cfv 5456   Topctop 16960   clsccl 17084   limPtclp 17200
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-top 16965  df-cld 17085  df-ntr 17086  df-cls 17087  df-nei 17164  df-lp 17202
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