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Theorem fileln0 22062
Description: An element of a filter is nonempty. (Contributed by FL, 24-May-2011.) (Revised by Mario Carneiro, 28-Jul-2015.)
Assertion
Ref Expression
fileln0 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝐴 ≠ ∅)

Proof of Theorem fileln0
StepHypRef Expression
1 id 22 . 2 (𝐴𝐹𝐴𝐹)
2 0nelfil 22061 . 2 (𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹)
3 nelne2 3068 . 2 ((𝐴𝐹 ∧ ¬ ∅ ∈ 𝐹) → 𝐴 ≠ ∅)
41, 2, 3syl2anr 590 1 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝐴 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386  wcel 2107  wne 2969  c0 4141  cfv 6135  Filcfil 22057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fv 6143  df-fbas 20139  df-fil 22058
This theorem is referenced by:  filinn0  22072  filintn0  22073  alexsublem  22256  cfil3i  23475  iscmet3  23499  filnetlem4  32964
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