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Theorem opnneiid 16863
Description: Only an open set is a neighborhood of itself. (Contributed by FL, 2-Oct-2006.)
Assertion
Ref Expression
opnneiid  |-  ( J  e.  Top  ->  ( N  e.  ( ( nei `  J ) `  N )  <->  N  e.  J ) )

Proof of Theorem opnneiid
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 neii2 16845 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  N ) )  ->  E. x  e.  J  ( N  C_  x  /\  x  C_  N ) )
2 eqss 3194 . . . . . 6  |-  ( N  =  x  <->  ( N  C_  x  /\  x  C_  N ) )
3 eleq1a 2352 . . . . . 6  |-  ( x  e.  J  ->  ( N  =  x  ->  N  e.  J ) )
42, 3syl5bir 209 . . . . 5  |-  ( x  e.  J  ->  (
( N  C_  x  /\  x  C_  N )  ->  N  e.  J
) )
54rexlimiv 2661 . . . 4  |-  ( E. x  e.  J  ( N  C_  x  /\  x  C_  N )  ->  N  e.  J )
61, 5syl 15 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  N ) )  ->  N  e.  J )
76ex 423 . 2  |-  ( J  e.  Top  ->  ( N  e.  ( ( nei `  J ) `  N )  ->  N  e.  J ) )
8 ssid 3197 . . 3  |-  N  C_  N
9 opnneiss 16855 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  J  /\  N  C_  N )  ->  N  e.  ( ( nei `  J ) `  N ) )
1093exp 1150 . . 3  |-  ( J  e.  Top  ->  ( N  e.  J  ->  ( N  C_  N  ->  N  e.  ( ( nei `  J ) `  N
) ) ) )
118, 10mpii 39 . 2  |-  ( J  e.  Top  ->  ( N  e.  J  ->  N  e.  ( ( nei `  J ) `  N
) ) )
127, 11impbid 183 1  |-  ( J  e.  Top  ->  ( N  e.  ( ( nei `  J ) `  N )  <->  N  e.  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   ` cfv 5255   Topctop 16631   neicnei 16834
This theorem is referenced by:  0nei  16865
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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