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Theorem islp2 16839
Description: The predicate " P is a limit point of  S," in terms of neighborhoods. Definition of limit point in [Munkres] p. 97. Although Munkres uses open neighborhoods, it also works for our more general neighborhoods. (Contributed by NM, 26-Feb-2007.) (Proof shortened by Mario Carneiro, 25-Dec-2016.)
Hypothesis
Ref Expression
lpfval.1  |-  X  = 
U. J
Assertion
Ref Expression
islp2  |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  X )  ->  ( P  e.  ( ( limPt `  J ) `  S )  <->  A. n  e.  ( ( nei `  J
) `  { P } ) ( n  i^i  ( S  \  { P } ) )  =/=  (/) ) )
Distinct variable groups:    n, J    P, n    S, n    n, X

Proof of Theorem islp2
StepHypRef Expression
1 lpfval.1 . . . 4  |-  X  = 
U. J
21islp 16834 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( limPt `  J ) `  S )  <->  P  e.  ( ( cls `  J
) `  ( S  \  { P } ) ) ) )
323adant3 980 . 2  |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  X )  ->  ( P  e.  ( ( limPt `  J ) `  S )  <->  P  e.  ( ( cls `  J
) `  ( S  \  { P } ) ) ) )
4 ssdifss 3282 . . 3  |-  ( S 
C_  X  ->  ( S  \  { P }
)  C_  X )
51neindisj2 16822 . . 3  |-  ( ( J  e.  Top  /\  ( S  \  { P } )  C_  X  /\  P  e.  X
)  ->  ( P  e.  ( ( cls `  J
) `  ( S  \  { P } ) )  <->  A. n  e.  ( ( nei `  J
) `  { P } ) ( n  i^i  ( S  \  { P } ) )  =/=  (/) ) )
64, 5syl3an2 1221 . 2  |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  X )  ->  ( P  e.  ( ( cls `  J ) `  ( S  \  { P } ) )  <->  A. n  e.  ( ( nei `  J
) `  { P } ) ( n  i^i  ( S  \  { P } ) )  =/=  (/) ) )
73, 6bitrd 246 1  |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  X )  ->  ( P  e.  ( ( limPt `  J ) `  S )  <->  A. n  e.  ( ( nei `  J
) `  { P } ) ( n  i^i  ( S  \  { P } ) )  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2421   A.wral 2518    \ cdif 3124    i^i cin 3126    C_ wss 3127   (/)c0 3430   {csn 3614   U.cuni 3801   ` cfv 4673   Topctop 16593   clsccl 16717   neicnei 16796   limPtclp 16828
This theorem is referenced by:  clslp  16841  lpbl  18011  reperflem  18285
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-top 16598  df-cld 16718  df-ntr 16719  df-cls 16720  df-nei 16797  df-lp 16830
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