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Theorem npmp 25521
Description: A neighborhood of a point can't be empty. (Contributed by FL, 15-Oct-2012.)
Assertion
Ref Expression
npmp  |-  ( ( J  e.  Top  /\  A  e.  X  /\  N  e.  ( ( nei `  J ) `  { A } ) )  ->  N  =/=  (/) )

Proof of Theorem npmp
StepHypRef Expression
1 simp1 955 . . 3  |-  ( ( J  e.  Top  /\  A  e.  X  /\  N  e.  ( ( nei `  J ) `  { A } ) )  ->  J  e.  Top )
2 snnzg 3743 . . . 4  |-  ( A  e.  X  ->  { A }  =/=  (/) )
323ad2ant2 977 . . 3  |-  ( ( J  e.  Top  /\  A  e.  X  /\  N  e.  ( ( nei `  J ) `  { A } ) )  ->  { A }  =/=  (/) )
4 0nnei 16849 . . 3  |-  ( ( J  e.  Top  /\  { A }  =/=  (/) )  ->  -.  (/)  e.  ( ( nei `  J ) `
 { A }
) )
51, 3, 4syl2anc 642 . 2  |-  ( ( J  e.  Top  /\  A  e.  X  /\  N  e.  ( ( nei `  J ) `  { A } ) )  ->  -.  (/)  e.  ( ( nei `  J
) `  { A } ) )
6 nelne2 2536 . . . 4  |-  ( ( N  e.  ( ( nei `  J ) `
 { A }
)  /\  -.  (/)  e.  ( ( nei `  J
) `  { A } ) )  ->  N  =/=  (/) )
76ex 423 . . 3  |-  ( N  e.  ( ( nei `  J ) `  { A } )  ->  ( -.  (/)  e.  ( ( nei `  J ) `
 { A }
)  ->  N  =/=  (/) ) )
873ad2ant3 978 . 2  |-  ( ( J  e.  Top  /\  A  e.  X  /\  N  e.  ( ( nei `  J ) `  { A } ) )  ->  ( -.  (/)  e.  ( ( nei `  J
) `  { A } )  ->  N  =/=  (/) ) )
95, 8mpd 14 1  |-  ( ( J  e.  Top  /\  A  e.  X  /\  N  e.  ( ( nei `  J ) `  { A } ) )  ->  N  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ w3a 934    e. wcel 1684    =/= wne 2446   (/)c0 3455   {csn 3640   ` 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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