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Theorem opnneissb 16851
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 16653 . . . . . 6  |-  ( ( J  e.  Top  /\  N  e.  J )  ->  N  C_  X )
32adantr 451 . . . . 5  |-  ( ( ( J  e.  Top  /\  N  e.  J )  /\  ( S  C_  X  /\  S  C_  N
) )  ->  N  C_  X )
4 ssid 3197 . . . . . . 7  |-  N  C_  N
5 sseq2 3200 . . . . . . . . 9  |-  ( g  =  N  ->  ( S  C_  g  <->  S  C_  N
) )
6 sseq1 3199 . . . . . . . . 9  |-  ( g  =  N  ->  (
g  C_  N  <->  N  C_  N
) )
75, 6anbi12d 691 . . . . . . . 8  |-  ( g  =  N  ->  (
( S  C_  g  /\  g  C_  N )  <-> 
( S  C_  N  /\  N  C_  N ) ) )
87rspcev 2884 . . . . . . 7  |-  ( ( N  e.  J  /\  ( S  C_  N  /\  N  C_  N ) )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
94, 8mpanr2 665 . . . . . 6  |-  ( ( N  e.  J  /\  S  C_  N )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
109ad2ant2l 726 . . . . 5  |-  ( ( ( J  e.  Top  /\  N  e.  J )  /\  ( S  C_  X  /\  S  C_  N
) )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
111isnei 16840 . . . . . 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 727 . . . . 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 888 . . . 4  |-  ( ( ( J  e.  Top  /\  N  e.  J )  /\  ( S  C_  X  /\  S  C_  N
) )  ->  N  e.  ( ( nei `  J
) `  S )
)
1413exp43 595 . . 3  |-  ( J  e.  Top  ->  ( N  e.  J  ->  ( S  C_  X  ->  ( S  C_  N  ->  N  e.  ( ( nei `  J ) `  S
) ) ) ) )
15143imp 1145 . 2  |-  ( ( J  e.  Top  /\  N  e.  J  /\  S  C_  X )  -> 
( S  C_  N  ->  N  e.  ( ( nei `  J ) `
 S ) ) )
16 ssnei 16847 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  N )
1716ex 423 . . 3  |-  ( J  e.  Top  ->  ( N  e.  ( ( nei `  J ) `  S )  ->  S  C_  N ) )
18173ad2ant1 976 . 2  |-  ( ( J  e.  Top  /\  N  e.  J  /\  S  C_  X )  -> 
( N  e.  ( ( nei `  J
) `  S )  ->  S  C_  N )
)
1915, 18impbid 183 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    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   U.cuni 3827   ` cfv 5255   Topctop 16631   neicnei 16834
This theorem is referenced by:  opnneiss  16855
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-top 16636  df-nei 16835
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