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Theorem opnssneib 12351
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 519 . . . . . 6  |-  ( ( ( S  e.  J  /\  N  C_  X )  /\  S  C_  N
)  ->  N  C_  X
)
2 sseq2 3121 . . . . . . . . . 10  |-  ( g  =  S  ->  ( S  C_  g  <->  S  C_  S
) )
3 sseq1 3120 . . . . . . . . . 10  |-  ( g  =  S  ->  (
g  C_  N  <->  S  C_  N
) )
42, 3anbi12d 464 . . . . . . . . 9  |-  ( g  =  S  ->  (
( S  C_  g  /\  g  C_  N )  <-> 
( S  C_  S  /\  S  C_  N ) ) )
5 ssid 3117 . . . . . . . . . 10  |-  S  C_  S
65biantrur 301 . . . . . . . . 9  |-  ( S 
C_  N  <->  ( S  C_  S  /\  S  C_  N ) )
74, 6syl6bbr 197 . . . . . . . 8  |-  ( g  =  S  ->  (
( S  C_  g  /\  g  C_  N )  <-> 
S  C_  N )
)
87rspcev 2789 . . . . . . 7  |-  ( ( S  e.  J  /\  S  C_  N )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
98adantlr 468 . . . . . 6  |-  ( ( ( S  e.  J  /\  N  C_  X )  /\  S  C_  N
)  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
101, 9jca 304 . . . . 5  |-  ( ( ( S  e.  J  /\  N  C_  X )  /\  S  C_  N
)  ->  ( N  C_  X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) )
1110ex 114 . . . 4  |-  ( ( S  e.  J  /\  N  C_  X )  -> 
( S  C_  N  ->  ( N  C_  X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) ) )
12113adant1 999 . . 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 12202 . . . . 5  |-  ( ( J  e.  Top  /\  S  e.  J )  ->  S  C_  X )
1513isnei 12339 . . . . 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 280 . . . 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 1001 . . 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 168 . 2  |-  ( ( J  e.  Top  /\  S  e.  J  /\  N  C_  X )  -> 
( S  C_  N  ->  N  e.  ( ( nei `  J ) `
 S ) ) )
19 ssnei 12346 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  N )
2019ex 114 . . 3  |-  ( J  e.  Top  ->  ( N  e.  ( ( nei `  J ) `  S )  ->  S  C_  N ) )
21203ad2ant1 1002 . 2  |-  ( ( J  e.  Top  /\  S  e.  J  /\  N  C_  X )  -> 
( N  e.  ( ( nei `  J
) `  S )  ->  S  C_  N )
)
2218, 21impbid 128 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 103    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480   E.wrex 2417    C_ wss 3071   U.cuni 3739   ` cfv 5126   Topctop 12190   neicnei 12333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4046  ax-sep 4049  ax-pow 4101  ax-pr 4134
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-iun 3818  df-br 3933  df-opab 3993  df-mpt 3994  df-id 4218  df-xp 4548  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-rn 4553  df-res 4554  df-ima 4555  df-iota 5091  df-fun 5128  df-fn 5129  df-f 5130  df-f1 5131  df-fo 5132  df-f1o 5133  df-fv 5134  df-top 12191  df-nei 12334
This theorem is referenced by:  neissex  12360
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