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

Theorem 0nelfil 23675
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 23674 . 2 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2 0nelfb 23657 . 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 2098  c0 4314  cfv 6533  fBascfbas 21216  Filcfil 23671
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fv 6541  df-fbas 21225  df-fil 23672
This theorem is referenced by:  fileln0  23676  isfil2  23682  infil  23689  filuni  23711  filufint  23746  rnelfmlem  23778  fmfnfm  23784  fclscmpi  23855
  Copyright terms: Public domain W3C validator