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Theorem neisspw 19901
Description: The neighborhoods of any set are subsets of the base set. (Contributed by Stefan O'Rear, 6-Aug-2015.)
Hypothesis
Ref Expression
neifval.1  |-  X  = 
U. J
Assertion
Ref Expression
neisspw  |-  ( J  e.  Top  ->  (
( nei `  J
) `  S )  C_ 
~P X )

Proof of Theorem neisspw
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 neifval.1 . . . . 5  |-  X  = 
U. J
21neii1 19900 . . . 4  |-  ( ( J  e.  Top  /\  v  e.  ( ( nei `  J ) `  S ) )  -> 
v  C_  X )
3 selpw 3962 . . . 4  |-  ( v  e.  ~P X  <->  v  C_  X )
42, 3sylibr 212 . . 3  |-  ( ( J  e.  Top  /\  v  e.  ( ( nei `  J ) `  S ) )  -> 
v  e.  ~P X
)
54ex 432 . 2  |-  ( J  e.  Top  ->  (
v  e.  ( ( nei `  J ) `
 S )  -> 
v  e.  ~P X
) )
65ssrdv 3448 1  |-  ( J  e.  Top  ->  (
( nei `  J
) `  S )  C_ 
~P X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    C_ wss 3414   ~Pcpw 3955   U.cuni 4191   ` cfv 5569   Topctop 19686   neicnei 19891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-top 19691  df-nei 19892
This theorem is referenced by:  hausflim  20774  flimclslem  20777  fclsfnflim  20820
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