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

Theorem 0nelfb 22005
Description: No filter base contains the empty set. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)
Assertion
Ref Expression
0nelfb (𝐹 ∈ (fBas‘𝐵) → ¬ ∅ ∈ 𝐹)

Proof of Theorem 0nelfb
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvdm 6465 . . . . 5 (𝐹 ∈ (fBas‘𝐵) → 𝐵 ∈ dom fBas)
2 isfbas 22003 . . . . 5 (𝐵 ∈ dom fBas → (𝐹 ∈ (fBas‘𝐵) ↔ (𝐹 ⊆ 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥𝐹𝑦𝐹 (𝐹 ∩ 𝒫 (𝑥𝑦)) ≠ ∅))))
31, 2syl 17 . . . 4 (𝐹 ∈ (fBas‘𝐵) → (𝐹 ∈ (fBas‘𝐵) ↔ (𝐹 ⊆ 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥𝐹𝑦𝐹 (𝐹 ∩ 𝒫 (𝑥𝑦)) ≠ ∅))))
43ibi 259 . . 3 (𝐹 ∈ (fBas‘𝐵) → (𝐹 ⊆ 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥𝐹𝑦𝐹 (𝐹 ∩ 𝒫 (𝑥𝑦)) ≠ ∅)))
5 simpr2 1256 . . 3 ((𝐹 ⊆ 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥𝐹𝑦𝐹 (𝐹 ∩ 𝒫 (𝑥𝑦)) ≠ ∅)) → ∅ ∉ 𝐹)
64, 5syl 17 . 2 (𝐹 ∈ (fBas‘𝐵) → ∅ ∉ 𝐹)
7 df-nel 3103 . 2 (∅ ∉ 𝐹 ↔ ¬ ∅ ∈ 𝐹)
86, 7sylib 210 1 (𝐹 ∈ (fBas‘𝐵) → ¬ ∅ ∈ 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  w3a 1113  wcel 2166  wne 2999  wnel 3102  wral 3117  cin 3797  wss 3798  c0 4144  𝒫 cpw 4378  dom cdm 5342  cfv 6123  fBascfbas 20094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fv 6131  df-fbas 20103
This theorem is referenced by:  fbdmn0  22008  fbncp  22013  fbun  22014  fbfinnfr  22015  0nelfil  22023  fsubbas  22041  fbasfip  22042  fgcl  22052  fbasrn  22058  uzfbas  22072  ucnextcn  22478
  Copyright terms: Public domain W3C validator