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Theorem opnssneib 17180
Description: Any superset of an open set is a neighborhood of it. (Contributed by NM, 14-Feb-2007.)
Hypothesis
Ref Expression
neips.1  |-  X  = 
U. J
Assertion
Ref Expression
opnssneib  |-  ( ( J  e.  Top  /\  S  e.  J  /\  N  C_  X )  -> 
( S  C_  N  <->  N  e.  ( ( nei `  J ) `  S
) ) )

Proof of Theorem opnssneib
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 simplr 733 . . . . . 6  |-  ( ( ( S  e.  J  /\  N  C_  X )  /\  S  C_  N
)  ->  N  C_  X
)
2 sseq2 3371 . . . . . . . . . 10  |-  ( g  =  S  ->  ( S  C_  g  <->  S  C_  S
) )
3 sseq1 3370 . . . . . . . . . 10  |-  ( g  =  S  ->  (
g  C_  N  <->  S  C_  N
) )
42, 3anbi12d 693 . . . . . . . . 9  |-  ( g  =  S  ->  (
( S  C_  g  /\  g  C_  N )  <-> 
( S  C_  S  /\  S  C_  N ) ) )
5 ssid 3368 . . . . . . . . . 10  |-  S  C_  S
65biantrur 494 . . . . . . . . 9  |-  ( S 
C_  N  <->  ( S  C_  S  /\  S  C_  N ) )
74, 6syl6bbr 256 . . . . . . . 8  |-  ( g  =  S  ->  (
( S  C_  g  /\  g  C_  N )  <-> 
S  C_  N )
)
87rspcev 3053 . . . . . . 7  |-  ( ( S  e.  J  /\  S  C_  N )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
98adantlr 697 . . . . . 6  |-  ( ( ( S  e.  J  /\  N  C_  X )  /\  S  C_  N
)  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
101, 9jca 520 . . . . 5  |-  ( ( ( S  e.  J  /\  N  C_  X )  /\  S  C_  N
)  ->  ( N  C_  X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) )
1110ex 425 . . . 4  |-  ( ( S  e.  J  /\  N  C_  X )  -> 
( S  C_  N  ->  ( N  C_  X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) ) )
12113adant1 976 . . 3  |-  ( ( J  e.  Top  /\  S  e.  J  /\  N  C_  X )  -> 
( S  C_  N  ->  ( N  C_  X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) ) )
13 neips.1 . . . . . 6  |-  X  = 
U. J
1413eltopss 16981 . . . . 5  |-  ( ( J  e.  Top  /\  S  e.  J )  ->  S  C_  X )
1513isnei 17168 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( N  e.  ( ( nei `  J
) `  S )  <->  ( N  C_  X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) ) )
1614, 15syldan 458 . . . 4  |-  ( ( J  e.  Top  /\  S  e.  J )  ->  ( N  e.  ( ( nei `  J
) `  S )  <->  ( N  C_  X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) ) )
17163adant3 978 . . 3  |-  ( ( J  e.  Top  /\  S  e.  J  /\  N  C_  X )  -> 
( N  e.  ( ( nei `  J
) `  S )  <->  ( N  C_  X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) ) )
1812, 17sylibrd 227 . 2  |-  ( ( J  e.  Top  /\  S  e.  J  /\  N  C_  X )  -> 
( S  C_  N  ->  N  e.  ( ( nei `  J ) `
 S ) ) )
19 ssnei 17175 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  N )
2019ex 425 . . 3  |-  ( J  e.  Top  ->  ( N  e.  ( ( nei `  J ) `  S )  ->  S  C_  N ) )
21203ad2ant1 979 . 2  |-  ( ( J  e.  Top  /\  S  e.  J  /\  N  C_  X )  -> 
( N  e.  ( ( nei `  J
) `  S )  ->  S  C_  N )
)
2218, 21impbid 185 1  |-  ( ( J  e.  Top  /\  S  e.  J  /\  N  C_  X )  -> 
( S  C_  N  <->  N  e.  ( ( nei `  J ) `  S
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E.wrex 2707    C_ wss 3321   U.cuni 4016   ` cfv 5455   Topctop 16959   neicnei 17162
This theorem is referenced by:  neissex  17192
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-top 16964  df-nei 17163
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