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Theorem opnssneib 13289
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 528 . . . . . 6  |-  ( ( ( S  e.  J  /\  N  C_  X )  /\  S  C_  N
)  ->  N  C_  X
)
2 sseq2 3179 . . . . . . . . . 10  |-  ( g  =  S  ->  ( S  C_  g  <->  S  C_  S
) )
3 sseq1 3178 . . . . . . . . . 10  |-  ( g  =  S  ->  (
g  C_  N  <->  S  C_  N
) )
42, 3anbi12d 473 . . . . . . . . 9  |-  ( g  =  S  ->  (
( S  C_  g  /\  g  C_  N )  <-> 
( S  C_  S  /\  S  C_  N ) ) )
5 ssid 3175 . . . . . . . . . 10  |-  S  C_  S
65biantrur 303 . . . . . . . . 9  |-  ( S 
C_  N  <->  ( S  C_  S  /\  S  C_  N ) )
74, 6bitr4di 198 . . . . . . . 8  |-  ( g  =  S  ->  (
( S  C_  g  /\  g  C_  N )  <-> 
S  C_  N )
)
87rspcev 2841 . . . . . . 7  |-  ( ( S  e.  J  /\  S  C_  N )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
98adantlr 477 . . . . . 6  |-  ( ( ( S  e.  J  /\  N  C_  X )  /\  S  C_  N
)  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
101, 9jca 306 . . . . 5  |-  ( ( ( S  e.  J  /\  N  C_  X )  /\  S  C_  N
)  ->  ( N  C_  X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) )
1110ex 115 . . . 4  |-  ( ( S  e.  J  /\  N  C_  X )  -> 
( S  C_  N  ->  ( N  C_  X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) ) )
12113adant1 1015 . . 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 13140 . . . . 5  |-  ( ( J  e.  Top  /\  S  e.  J )  ->  S  C_  X )
1513isnei 13277 . . . . 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 282 . . . 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 1017 . . 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 169 . 2  |-  ( ( J  e.  Top  /\  S  e.  J  /\  N  C_  X )  -> 
( S  C_  N  ->  N  e.  ( ( nei `  J ) `
 S ) ) )
19 ssnei 13284 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  N )
2019ex 115 . . 3  |-  ( J  e.  Top  ->  ( N  e.  ( ( nei `  J ) `  S )  ->  S  C_  N ) )
21203ad2ant1 1018 . 2  |-  ( ( J  e.  Top  /\  S  e.  J  /\  N  C_  X )  -> 
( N  e.  ( ( nei `  J
) `  S )  ->  S  C_  N )
)
2218, 21impbid 129 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    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   E.wrex 2456    C_ wss 3129   U.cuni 3807   ` cfv 5211   Topctop 13128   neicnei 13271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-top 13129  df-nei 13272
This theorem is referenced by:  neissex  13298
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