Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  0nelfil Structured version   Visualization version   GIF version

Theorem 0nelfil 22549
 Description: The empty set doesn't belong to a filter. (Contributed by FL, 20-Jul-2007.) (Revised by Mario Carneiro, 28-Jul-2015.)
Assertion
Ref Expression
0nelfil (𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹)

Proof of Theorem 0nelfil
StepHypRef Expression
1 filfbas 22548 . 2 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2 0nelfb 22531 . 2 (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐹)
31, 2syl 17 1 (𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2111  ∅c0 4225  ‘cfv 6335  fBascfbas 20154  Filcfil 22545 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fv 6343  df-fbas 20163  df-fil 22546 This theorem is referenced by:  fileln0  22550  isfil2  22556  infil  22563  filuni  22585  filufint  22620  rnelfmlem  22652  fmfnfm  22658  fclscmpi  22729
 Copyright terms: Public domain W3C validator