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Theorem islpi 19945
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 19944 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  ( S  u.  ( ( limPt `  J
) `  S )
) )
32eleq2d 2474 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( cls `  J
) `  S )  <->  P  e.  ( S  u.  ( ( limPt `  J
) `  S )
) ) )
4 elun 3586 . . . . 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 251 . . . 4  |-  ( P  e.  ( S  u.  ( ( limPt `  J
) `  S )
)  <->  ( -.  P  e.  S  ->  P  e.  ( ( limPt `  J
) `  S )
) )
73, 6syl6bb 263 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( cls `  J
) `  S )  <->  ( -.  P  e.  S  ->  P  e.  ( (
limPt `  J ) `  S ) ) ) )
87biimpd 209 . 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 1407    e. wcel 1844    u. cun 3414    C_ wss 3416   U.cuni 4193   ` cfv 5571   Topctop 19688   clsccl 19813   limPtclp 19930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-iin 4276  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-top 19693  df-cld 19814  df-ntr 19815  df-cls 19816  df-nei 19894  df-lp 19932
This theorem is referenced by: (None)
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