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Theorem opnneissb 17094
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 16896 . . . . . 6  |-  ( ( J  e.  Top  /\  N  e.  J )  ->  N  C_  X )
32adantr 452 . . . . 5  |-  ( ( ( J  e.  Top  /\  N  e.  J )  /\  ( S  C_  X  /\  S  C_  N
) )  ->  N  C_  X )
4 ssid 3303 . . . . . . 7  |-  N  C_  N
5 sseq2 3306 . . . . . . . . 9  |-  ( g  =  N  ->  ( S  C_  g  <->  S  C_  N
) )
6 sseq1 3305 . . . . . . . . 9  |-  ( g  =  N  ->  (
g  C_  N  <->  N  C_  N
) )
75, 6anbi12d 692 . . . . . . . 8  |-  ( g  =  N  ->  (
( S  C_  g  /\  g  C_  N )  <-> 
( S  C_  N  /\  N  C_  N ) ) )
87rspcev 2988 . . . . . . 7  |-  ( ( N  e.  J  /\  ( S  C_  N  /\  N  C_  N ) )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
94, 8mpanr2 666 . . . . . 6  |-  ( ( N  e.  J  /\  S  C_  N )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
109ad2ant2l 727 . . . . 5  |-  ( ( ( J  e.  Top  /\  N  e.  J )  /\  ( S  C_  X  /\  S  C_  N
) )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
111isnei 17083 . . . . . 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 728 . . . . 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 889 . . . 4  |-  ( ( ( J  e.  Top  /\  N  e.  J )  /\  ( S  C_  X  /\  S  C_  N
) )  ->  N  e.  ( ( nei `  J
) `  S )
)
1413exp43 596 . . 3  |-  ( J  e.  Top  ->  ( N  e.  J  ->  ( S  C_  X  ->  ( S  C_  N  ->  N  e.  ( ( nei `  J ) `  S
) ) ) ) )
15143imp 1147 . 2  |-  ( ( J  e.  Top  /\  N  e.  J  /\  S  C_  X )  -> 
( S  C_  N  ->  N  e.  ( ( nei `  J ) `
 S ) ) )
16 ssnei 17090 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  N )
1716ex 424 . . 3  |-  ( J  e.  Top  ->  ( N  e.  ( ( nei `  J ) `  S )  ->  S  C_  N ) )
18173ad2ant1 978 . 2  |-  ( ( J  e.  Top  /\  N  e.  J  /\  S  C_  X )  -> 
( N  e.  ( ( nei `  J
) `  S )  ->  S  C_  N )
)
1915, 18impbid 184 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   E.wrex 2643    C_ wss 3256   U.cuni 3950   ` cfv 5387   Topctop 16874   neicnei 17077
This theorem is referenced by:  opnneiss  17098
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-top 16879  df-nei 17078
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