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Theorem opnssneib 19783
Description: Any superset of an open set is a neighborhood of it. (Contributed by NM, 14-Feb-2007.)
Hypothesis
Ref Expression
neips.1  |-  X  = 
U. J
Assertion
Ref Expression
opnssneib  |-  ( ( J  e.  Top  /\  S  e.  J  /\  N  C_  X )  -> 
( S  C_  N  <->  N  e.  ( ( nei `  J ) `  S
) ) )

Proof of Theorem opnssneib
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 simplr 753 . . . . . 6  |-  ( ( ( S  e.  J  /\  N  C_  X )  /\  S  C_  N
)  ->  N  C_  X
)
2 sseq2 3511 . . . . . . . . . 10  |-  ( g  =  S  ->  ( S  C_  g  <->  S  C_  S
) )
3 sseq1 3510 . . . . . . . . . 10  |-  ( g  =  S  ->  (
g  C_  N  <->  S  C_  N
) )
42, 3anbi12d 708 . . . . . . . . 9  |-  ( g  =  S  ->  (
( S  C_  g  /\  g  C_  N )  <-> 
( S  C_  S  /\  S  C_  N ) ) )
5 ssid 3508 . . . . . . . . . 10  |-  S  C_  S
65biantrur 504 . . . . . . . . 9  |-  ( S 
C_  N  <->  ( S  C_  S  /\  S  C_  N ) )
74, 6syl6bbr 263 . . . . . . . 8  |-  ( g  =  S  ->  (
( S  C_  g  /\  g  C_  N )  <-> 
S  C_  N )
)
87rspcev 3207 . . . . . . 7  |-  ( ( S  e.  J  /\  S  C_  N )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
98adantlr 712 . . . . . 6  |-  ( ( ( S  e.  J  /\  N  C_  X )  /\  S  C_  N
)  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
101, 9jca 530 . . . . 5  |-  ( ( ( S  e.  J  /\  N  C_  X )  /\  S  C_  N
)  ->  ( N  C_  X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) )
1110ex 432 . . . 4  |-  ( ( S  e.  J  /\  N  C_  X )  -> 
( S  C_  N  ->  ( N  C_  X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) ) )
12113adant1 1012 . . 3  |-  ( ( J  e.  Top  /\  S  e.  J  /\  N  C_  X )  -> 
( S  C_  N  ->  ( N  C_  X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) ) )
13 neips.1 . . . . . 6  |-  X  = 
U. J
1413eltopss 19583 . . . . 5  |-  ( ( J  e.  Top  /\  S  e.  J )  ->  S  C_  X )
1513isnei 19771 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( N  e.  ( ( nei `  J
) `  S )  <->  ( N  C_  X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) ) )
1614, 15syldan 468 . . . 4  |-  ( ( J  e.  Top  /\  S  e.  J )  ->  ( N  e.  ( ( nei `  J
) `  S )  <->  ( N  C_  X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) ) )
17163adant3 1014 . . 3  |-  ( ( J  e.  Top  /\  S  e.  J  /\  N  C_  X )  -> 
( N  e.  ( ( nei `  J
) `  S )  <->  ( N  C_  X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) ) )
1812, 17sylibrd 234 . 2  |-  ( ( J  e.  Top  /\  S  e.  J  /\  N  C_  X )  -> 
( S  C_  N  ->  N  e.  ( ( nei `  J ) `
 S ) ) )
19 ssnei 19778 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  N )
2019ex 432 . . 3  |-  ( J  e.  Top  ->  ( N  e.  ( ( nei `  J ) `  S )  ->  S  C_  N ) )
21203ad2ant1 1015 . 2  |-  ( ( J  e.  Top  /\  S  e.  J  /\  N  C_  X )  -> 
( N  e.  ( ( nei `  J
) `  S )  ->  S  C_  N )
)
2218, 21impbid 191 1  |-  ( ( J  e.  Top  /\  S  e.  J  /\  N  C_  X )  -> 
( S  C_  N  <->  N  e.  ( ( nei `  J ) `  S
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   E.wrex 2805    C_ wss 3461   U.cuni 4235   ` cfv 5570   Topctop 19561   neicnei 19765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-top 19566  df-nei 19766
This theorem is referenced by:  neissex  19795
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