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Theorem opnneissb 16867
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 16669 . . . . . 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 3210 . . . . . . 7  |-  N  C_  N
5 sseq2 3213 . . . . . . . . 9  |-  ( g  =  N  ->  ( S  C_  g  <->  S  C_  N
) )
6 sseq1 3212 . . . . . . . . 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 2897 . . . . . . 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 16856 . . . . . 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 16863 . . . 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 1632    e. wcel 1696   E.wrex 2557    C_ wss 3165   U.cuni 3843   ` cfv 5271   Topctop 16647   neicnei 16850
This theorem is referenced by:  opnneiss  16871
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-top 16652  df-nei 16851
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