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Theorem lpbl 18065
Description: Every ball around a limit point  P of a subset  S includes a member of  S (even if  P  e/  S). (Contributed by NM, 9-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.)
Hypothesis
Ref Expression
mopni.1  |-  J  =  ( MetOpen `  D )
Assertion
Ref Expression
lpbl  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  E. x  e.  S  x  e.  ( P
( ball `  D ) R ) )
Distinct variable groups:    x, D    x, J    x, R    x, S    x, P    x, X

Proof of Theorem lpbl
StepHypRef Expression
1 simpl1 958 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  D  e.  ( * Met `  X ) )
2 mopni.1 . . . . . . . . 9  |-  J  =  ( MetOpen `  D )
32mopntop 18002 . . . . . . . 8  |-  ( D  e.  ( * Met `  X )  ->  J  e.  Top )
41, 3syl 15 . . . . . . 7  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  J  e.  Top )
5 simpl2 959 . . . . . . . 8  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  S  C_  X )
62mopnuni 18003 . . . . . . . . 9  |-  ( D  e.  ( * Met `  X )  ->  X  =  U. J )
71, 6syl 15 . . . . . . . 8  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  X  =  U. J )
85, 7sseqtrd 3227 . . . . . . 7  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  S  C_  U. J )
9 eqid 2296 . . . . . . . 8  |-  U. J  =  U. J
109lpss 16890 . . . . . . 7  |-  ( ( J  e.  Top  /\  S  C_  U. J )  ->  ( ( limPt `  J ) `  S
)  C_  U. J )
114, 8, 10syl2anc 642 . . . . . 6  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  -> 
( ( limPt `  J
) `  S )  C_ 
U. J )
12 simpl3 960 . . . . . 6  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  P  e.  ( ( limPt `  J ) `  S ) )
1311, 12sseldd 3194 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  P  e.  U. J )
1413, 7eleqtrrd 2373 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  P  e.  X )
15 simpr 447 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  R  e.  RR+ )
162blnei 18064 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR+ )  ->  ( P ( ball `  D ) R )  e.  ( ( nei `  J ) `  { P } ) )
171, 14, 15, 16syl3anc 1182 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  -> 
( P ( ball `  D ) R )  e.  ( ( nei `  J ) `  { P } ) )
189islp2 16893 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  U. J  /\  P  e.  U. J )  ->  ( P  e.  ( ( limPt `  J
) `  S )  <->  A. x  e.  ( ( nei `  J ) `
 { P }
) ( x  i^i  ( S  \  { P } ) )  =/=  (/) ) )
194, 8, 13, 18syl3anc 1182 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  -> 
( P  e.  ( ( limPt `  J ) `  S )  <->  A. x  e.  ( ( nei `  J
) `  { P } ) ( x  i^i  ( S  \  { P } ) )  =/=  (/) ) )
2012, 19mpbid 201 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  A. x  e.  (
( nei `  J
) `  { P } ) ( x  i^i  ( S  \  { P } ) )  =/=  (/) )
21 ineq1 3376 . . . . 5  |-  ( x  =  ( P (
ball `  D ) R )  ->  (
x  i^i  ( S  \  { P } ) )  =  ( ( P ( ball `  D
) R )  i^i  ( S  \  { P } ) ) )
2221neeq1d 2472 . . . 4  |-  ( x  =  ( P (
ball `  D ) R )  ->  (
( x  i^i  ( S  \  { P }
) )  =/=  (/)  <->  ( ( P ( ball `  D
) R )  i^i  ( S  \  { P } ) )  =/=  (/) ) )
2322rspcva 2895 . . 3  |-  ( ( ( P ( ball `  D ) R )  e.  ( ( nei `  J ) `  { P } )  /\  A. x  e.  ( ( nei `  J ) `  { P } ) ( x  i^i  ( S 
\  { P }
) )  =/=  (/) )  -> 
( ( P (
ball `  D ) R )  i^i  ( S  \  { P }
) )  =/=  (/) )
2417, 20, 23syl2anc 642 . 2  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  -> 
( ( P (
ball `  D ) R )  i^i  ( S  \  { P }
) )  =/=  (/) )
25 elin 3371 . . . . 5  |-  ( x  e.  ( ( P ( ball `  D
) R )  i^i  ( S  \  { P } ) )  <->  ( x  e.  ( P ( ball `  D ) R )  /\  x  e.  ( S  \  { P } ) ) )
26 eldifi 3311 . . . . . . 7  |-  ( x  e.  ( S  \  { P } )  ->  x  e.  S )
2726anim2i 552 . . . . . 6  |-  ( ( x  e.  ( P ( ball `  D
) R )  /\  x  e.  ( S  \  { P } ) )  ->  ( x  e.  ( P ( ball `  D ) R )  /\  x  e.  S
) )
2827ancomd 438 . . . . 5  |-  ( ( x  e.  ( P ( ball `  D
) R )  /\  x  e.  ( S  \  { P } ) )  ->  ( x  e.  S  /\  x  e.  ( P ( ball `  D ) R ) ) )
2925, 28sylbi 187 . . . 4  |-  ( x  e.  ( ( P ( ball `  D
) R )  i^i  ( S  \  { P } ) )  -> 
( x  e.  S  /\  x  e.  ( P ( ball `  D
) R ) ) )
3029eximi 1566 . . 3  |-  ( E. x  x  e.  ( ( P ( ball `  D ) R )  i^i  ( S  \  { P } ) )  ->  E. x ( x  e.  S  /\  x  e.  ( P ( ball `  D ) R ) ) )
31 n0 3477 . . 3  |-  ( ( ( P ( ball `  D ) R )  i^i  ( S  \  { P } ) )  =/=  (/)  <->  E. x  x  e.  ( ( P (
ball `  D ) R )  i^i  ( S  \  { P }
) ) )
32 df-rex 2562 . . 3  |-  ( E. x  e.  S  x  e.  ( P (
ball `  D ) R )  <->  E. x
( x  e.  S  /\  x  e.  ( P ( ball `  D
) R ) ) )
3330, 31, 323imtr4i 257 . 2  |-  ( ( ( P ( ball `  D ) R )  i^i  ( S  \  { P } ) )  =/=  (/)  ->  E. x  e.  S  x  e.  ( P ( ball `  D
) R ) )
3424, 33syl 15 1  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  E. x  e.  S  x  e.  ( P
( ball `  D ) R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468   {csn 3653   U.cuni 3843   ` cfv 5271  (class class class)co 5874   RR+crp 10370   * Metcxmt 16385   ballcbl 16387   MetOpencmopn 16388   Topctop 16647   neicnei 16850   limPtclp 16882
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  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  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-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  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-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  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-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-topgen 13360  df-xmet 16389  df-bl 16391  df-mopn 16392  df-top 16652  df-bases 16654  df-topon 16655  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884
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