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

Theorem filn0 22386
Description: The empty set is not a filter. Remark below Definition 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 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)

Proof of Theorem filn0
StepHypRef Expression
1 filtop 22379 . 2 (𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)
21ne0d 4304 1 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wne 3020  c0 4294  cfv 6351  Filcfil 22369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fv 6359  df-fbas 20458  df-fil 22370
This theorem is referenced by:  ufileu  22443  filufint  22444  uffixfr  22447  uffix2  22448  uffixsn  22449  hausflim  22505  fclsval  22532  isfcls  22533  fclsopn  22538  fclsfnflim  22551  filnetlem4  33614
  Copyright terms: Public domain W3C validator