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Theorem opnneiid 19754
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 19736 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  N ) )  ->  E. x  e.  J  ( N  C_  x  /\  x  C_  N ) )
2 eqss 3514 . . . . . 6  |-  ( N  =  x  <->  ( N  C_  x  /\  x  C_  N ) )
3 eleq1a 2540 . . . . . 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 2943 . . . 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 3518 . . 3  |-  N  C_  N
9 opnneiss 19746 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  J  /\  N  C_  N )  ->  N  e.  ( ( nei `  J ) `  N ) )
1093exp 1195 . . 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 1395    e. wcel 1819   E.wrex 2808    C_ wss 3471   ` cfv 5594   Topctop 19521   neicnei 19725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-top 19526  df-nei 19726
This theorem is referenced by:  0nei  19756
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