MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  filn0 Unicode version

Theorem filn0 17877
Description: The empty set is not a filter. Remark below def. 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 30-Oct-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
Assertion
Ref Expression
filn0  |-  ( F  e.  ( Fil `  X
)  ->  F  =/=  (/) )

Proof of Theorem filn0
StepHypRef Expression
1 filtop 17870 . 2  |-  ( F  e.  ( Fil `  X
)  ->  X  e.  F )
2 ne0i 3621 . 2  |-  ( X  e.  F  ->  F  =/=  (/) )
31, 2syl 16 1  |-  ( F  e.  ( Fil `  X
)  ->  F  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725    =/= wne 2593   (/)c0 3615   ` cfv 5440   Filcfil 17860
This theorem is referenced by:  ufileu  17934  filufint  17935  uffixfr  17938  uffix2  17939  uffixsn  17940  hausflim  17996  fclsval  18023  isfcls  18024  fclsopn  18029  fclsfnflim  18042  filnetlem4  26342
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fv 5448  df-fbas 16682  df-fil 17861
  Copyright terms: Public domain W3C validator