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Theorem apnei 25520
Description: Any point has a neighborhood. (Contributed by FL, 15-Oct-2012.)
Hypothesis
Ref Expression
apnei.1  |-  X  = 
U. J
Assertion
Ref Expression
apnei  |-  ( ( J  e.  Top  /\  A  e.  X )  ->  E. v  v  e.  ( ( nei `  J
) `  { A } ) )
Distinct variable groups:    v, A    v, J    v, X

Proof of Theorem apnei
StepHypRef Expression
1 snssi 3759 . . . . 5  |-  ( A  e.  X  ->  { A }  C_  X )
2 apnei.1 . . . . . 6  |-  X  = 
U. J
32tpnei 16858 . . . . 5  |-  ( J  e.  Top  ->  ( { A }  C_  X  <->  X  e.  ( ( nei `  J ) `  { A } ) ) )
41, 3syl5ibcom 211 . . . 4  |-  ( A  e.  X  ->  ( J  e.  Top  ->  X  e.  ( ( nei `  J
) `  { A } ) ) )
54impcom 419 . . 3  |-  ( ( J  e.  Top  /\  A  e.  X )  ->  X  e.  ( ( nei `  J ) `
 { A }
) )
6 eleq1 2343 . . . 4  |-  ( v  =  X  ->  (
v  e.  ( ( nei `  J ) `
 { A }
)  <->  X  e.  (
( nei `  J
) `  { A } ) ) )
76rspcev 2884 . . 3  |-  ( ( X  e.  ( ( nei `  J ) `
 { A }
)  /\  X  e.  ( ( nei `  J
) `  { A } ) )  ->  E. v  e.  (
( nei `  J
) `  { A } ) v  e.  ( ( nei `  J
) `  { A } ) )
85, 5, 7syl2anc 642 . 2  |-  ( ( J  e.  Top  /\  A  e.  X )  ->  E. v  e.  ( ( nei `  J
) `  { A } ) v  e.  ( ( nei `  J
) `  { A } ) )
9 df-rex 2549 . . 3  |-  ( E. v  e.  ( ( nei `  J ) `
 { A }
) v  e.  ( ( nei `  J
) `  { A } )  <->  E. v
( v  e.  ( ( nei `  J
) `  { A } )  /\  v  e.  ( ( nei `  J
) `  { A } ) ) )
10 anidm 625 . . . 4  |-  ( ( v  e.  ( ( nei `  J ) `
 { A }
)  /\  v  e.  ( ( nei `  J
) `  { A } ) )  <->  v  e.  ( ( nei `  J
) `  { A } ) )
1110exbii 1569 . . 3  |-  ( E. v ( v  e.  ( ( nei `  J
) `  { A } )  /\  v  e.  ( ( nei `  J
) `  { A } ) )  <->  E. v 
v  e.  ( ( nei `  J ) `
 { A }
) )
129, 11bitri 240 . 2  |-  ( E. v  e.  ( ( nei `  J ) `
 { A }
) v  e.  ( ( nei `  J
) `  { A } )  <->  E. v 
v  e.  ( ( nei `  J ) `
 { A }
) )
138, 12sylib 188 1  |-  ( ( J  e.  Top  /\  A  e.  X )  ->  E. v  v  e.  ( ( nei `  J
) `  { A } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   {csn 3640   U.cuni 3827   ` cfv 5255   Topctop 16631   neicnei 16834
This theorem is referenced by:  limptlimpr2lem1  25574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-top 16636  df-nei 16835
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