MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  neisspw Structured version   Unicode version

Theorem neisspw 19367
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 19366 . . . 4  |-  ( ( J  e.  Top  /\  v  e.  ( ( nei `  J ) `  S ) )  -> 
v  C_  X )
3 selpw 4010 . . . 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 434 . 2  |-  ( J  e.  Top  ->  (
v  e.  ( ( nei `  J ) `
 S )  -> 
v  e.  ~P X
) )
65ssrdv 3503 1  |-  ( J  e.  Top  ->  (
( nei `  J
) `  S )  C_ 
~P X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762    C_ wss 3469   ~Pcpw 4003   U.cuni 4238   ` cfv 5579   Topctop 19154   neicnei 19357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-top 19159  df-nei 19358
This theorem is referenced by:  hausflim  20210  flimclslem  20213  fclsfnflim  20256
  Copyright terms: Public domain W3C validator