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Theorem opnneissb 14107
Description: An open set is a neighborhood of any of its subsets. (Contributed by FL, 2-Oct-2006.)
Hypothesis
Ref Expression
neips.1  |-  X  = 
U. J
Assertion
Ref Expression
opnneissb  |-  ( ( J  e.  Top  /\  N  e.  J  /\  S  C_  X )  -> 
( S  C_  N  <->  N  e.  ( ( nei `  J ) `  S
) ) )

Proof of Theorem opnneissb
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 neips.1 . . . . . . 7  |-  X  = 
U. J
21eltopss 13961 . . . . . 6  |-  ( ( J  e.  Top  /\  N  e.  J )  ->  N  C_  X )
32adantr 276 . . . . 5  |-  ( ( ( J  e.  Top  /\  N  e.  J )  /\  ( S  C_  X  /\  S  C_  N
) )  ->  N  C_  X )
4 ssid 3190 . . . . . . 7  |-  N  C_  N
5 sseq2 3194 . . . . . . . . 9  |-  ( g  =  N  ->  ( S  C_  g  <->  S  C_  N
) )
6 sseq1 3193 . . . . . . . . 9  |-  ( g  =  N  ->  (
g  C_  N  <->  N  C_  N
) )
75, 6anbi12d 473 . . . . . . . 8  |-  ( g  =  N  ->  (
( S  C_  g  /\  g  C_  N )  <-> 
( S  C_  N  /\  N  C_  N ) ) )
87rspcev 2856 . . . . . . 7  |-  ( ( N  e.  J  /\  ( S  C_  N  /\  N  C_  N ) )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
94, 8mpanr2 438 . . . . . 6  |-  ( ( N  e.  J  /\  S  C_  N )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
109ad2ant2l 508 . . . . 5  |-  ( ( ( J  e.  Top  /\  N  e.  J )  /\  ( S  C_  X  /\  S  C_  N
) )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
111isnei 14096 . . . . . 6  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( N  e.  ( ( nei `  J
) `  S )  <->  ( N  C_  X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) ) )
1211ad2ant2r 509 . . . . 5  |-  ( ( ( J  e.  Top  /\  N  e.  J )  /\  ( S  C_  X  /\  S  C_  N
) )  ->  ( N  e.  ( ( nei `  J ) `  S )  <->  ( N  C_  X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) ) )
133, 10, 12mpbir2and 946 . . . 4  |-  ( ( ( J  e.  Top  /\  N  e.  J )  /\  ( S  C_  X  /\  S  C_  N
) )  ->  N  e.  ( ( nei `  J
) `  S )
)
1413exp43 372 . . 3  |-  ( J  e.  Top  ->  ( N  e.  J  ->  ( S  C_  X  ->  ( S  C_  N  ->  N  e.  ( ( nei `  J ) `  S
) ) ) ) )
15143imp 1195 . 2  |-  ( ( J  e.  Top  /\  N  e.  J  /\  S  C_  X )  -> 
( S  C_  N  ->  N  e.  ( ( nei `  J ) `
 S ) ) )
16 ssnei 14103 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  N )
1716ex 115 . . 3  |-  ( J  e.  Top  ->  ( N  e.  ( ( nei `  J ) `  S )  ->  S  C_  N ) )
18173ad2ant1 1020 . 2  |-  ( ( J  e.  Top  /\  N  e.  J  /\  S  C_  X )  -> 
( N  e.  ( ( nei `  J
) `  S )  ->  S  C_  N )
)
1915, 18impbid 129 1  |-  ( ( J  e.  Top  /\  N  e.  J  /\  S  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 980    = wceq 1364    e. wcel 2160   E.wrex 2469    C_ wss 3144   U.cuni 3824   ` cfv 5235   Topctop 13949   neicnei 14090
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-top 13950  df-nei 14091
This theorem is referenced by:  opnneiss  14110
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