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Theorem opnneiid 19386
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 19368 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  N ) )  ->  E. x  e.  J  ( N  C_  x  /\  x  C_  N ) )
2 eqss 3512 . . . . . 6  |-  ( N  =  x  <->  ( N  C_  x  /\  x  C_  N ) )
3 eleq1a 2543 . . . . . 6  |-  ( x  e.  J  ->  ( N  =  x  ->  N  e.  J ) )
42, 3syl5bir 218 . . . . 5  |-  ( x  e.  J  ->  (
( N  C_  x  /\  x  C_  N )  ->  N  e.  J
) )
54rexlimiv 2942 . . . 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 434 . 2  |-  ( J  e.  Top  ->  ( N  e.  ( ( nei `  J ) `  N )  ->  N  e.  J ) )
8 ssid 3516 . . 3  |-  N  C_  N
9 opnneiss 19378 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  J  /\  N  C_  N )  ->  N  e.  ( ( nei `  J ) `  N ) )
1093exp 1190 . . 3  |-  ( J  e.  Top  ->  ( N  e.  J  ->  ( N  C_  N  ->  N  e.  ( ( nei `  J ) `  N
) ) ) )
118, 10mpii 43 . 2  |-  ( J  e.  Top  ->  ( N  e.  J  ->  N  e.  ( ( nei `  J ) `  N
) ) )
127, 11impbid 191 1  |-  ( J  e.  Top  ->  ( N  e.  ( ( nei `  J ) `  N )  <->  N  e.  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   E.wrex 2808    C_ wss 3469   ` cfv 5579   Topctop 19154   neicnei 19357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-top 19159  df-nei 19358
This theorem is referenced by:  0nei  19388
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