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Theorem opnneiid 17114
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 17096 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  N ) )  ->  E. x  e.  J  ( N  C_  x  /\  x  C_  N ) )
2 eqss 3307 . . . . . 6  |-  ( N  =  x  <->  ( N  C_  x  /\  x  C_  N ) )
3 eleq1a 2457 . . . . . 6  |-  ( x  e.  J  ->  ( N  =  x  ->  N  e.  J ) )
42, 3syl5bir 210 . . . . 5  |-  ( x  e.  J  ->  (
( N  C_  x  /\  x  C_  N )  ->  N  e.  J
) )
54rexlimiv 2768 . . . 4  |-  ( E. x  e.  J  ( N  C_  x  /\  x  C_  N )  ->  N  e.  J )
61, 5syl 16 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  N ) )  ->  N  e.  J )
76ex 424 . 2  |-  ( J  e.  Top  ->  ( N  e.  ( ( nei `  J ) `  N )  ->  N  e.  J ) )
8 ssid 3311 . . 3  |-  N  C_  N
9 opnneiss 17106 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  J  /\  N  C_  N )  ->  N  e.  ( ( nei `  J ) `  N ) )
1093exp 1152 . . 3  |-  ( J  e.  Top  ->  ( N  e.  J  ->  ( N  C_  N  ->  N  e.  ( ( nei `  J ) `  N
) ) ) )
118, 10mpii 41 . 2  |-  ( J  e.  Top  ->  ( N  e.  J  ->  N  e.  ( ( nei `  J ) `  N
) ) )
127, 11impbid 184 1  |-  ( J  e.  Top  ->  ( N  e.  ( ( nei `  J ) `  N )  <->  N  e.  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   E.wrex 2651    C_ wss 3264   ` cfv 5395   Topctop 16882   neicnei 17085
This theorem is referenced by:  0nei  17116
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-top 16887  df-nei 17086
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