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Theorem opnneissb 12333
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 12185 . . . . . 6  |-  ( ( J  e.  Top  /\  N  e.  J )  ->  N  C_  X )
32adantr 274 . . . . 5  |-  ( ( ( J  e.  Top  /\  N  e.  J )  /\  ( S  C_  X  /\  S  C_  N
) )  ->  N  C_  X )
4 ssid 3117 . . . . . . 7  |-  N  C_  N
5 sseq2 3121 . . . . . . . . 9  |-  ( g  =  N  ->  ( S  C_  g  <->  S  C_  N
) )
6 sseq1 3120 . . . . . . . . 9  |-  ( g  =  N  ->  (
g  C_  N  <->  N  C_  N
) )
75, 6anbi12d 464 . . . . . . . 8  |-  ( g  =  N  ->  (
( S  C_  g  /\  g  C_  N )  <-> 
( S  C_  N  /\  N  C_  N ) ) )
87rspcev 2789 . . . . . . 7  |-  ( ( N  e.  J  /\  ( S  C_  N  /\  N  C_  N ) )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
94, 8mpanr2 434 . . . . . 6  |-  ( ( N  e.  J  /\  S  C_  N )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
109ad2ant2l 499 . . . . 5  |-  ( ( ( J  e.  Top  /\  N  e.  J )  /\  ( S  C_  X  /\  S  C_  N
) )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
111isnei 12322 . . . . . 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 500 . . . . 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 928 . . . 4  |-  ( ( ( J  e.  Top  /\  N  e.  J )  /\  ( S  C_  X  /\  S  C_  N
) )  ->  N  e.  ( ( nei `  J
) `  S )
)
1413exp43 369 . . 3  |-  ( J  e.  Top  ->  ( N  e.  J  ->  ( S  C_  X  ->  ( S  C_  N  ->  N  e.  ( ( nei `  J ) `  S
) ) ) ) )
15143imp 1175 . 2  |-  ( ( J  e.  Top  /\  N  e.  J  /\  S  C_  X )  -> 
( S  C_  N  ->  N  e.  ( ( nei `  J ) `
 S ) ) )
16 ssnei 12329 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  N )
1716ex 114 . . 3  |-  ( J  e.  Top  ->  ( N  e.  ( ( nei `  J ) `  S )  ->  S  C_  N ) )
18173ad2ant1 1002 . 2  |-  ( ( J  e.  Top  /\  N  e.  J  /\  S  C_  X )  -> 
( N  e.  ( ( nei `  J
) `  S )  ->  S  C_  N )
)
1915, 18impbid 128 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 103    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480   E.wrex 2417    C_ wss 3071   U.cuni 3736   ` cfv 5123   Topctop 12173   neicnei 12316
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 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131
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 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-top 12174  df-nei 12317
This theorem is referenced by:  opnneiss  12336
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