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Theorem filn0 22562
 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 22555 . 2 (𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)
21ne0d 4234 1 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2111   ≠ wne 2951  ∅c0 4225  ‘cfv 6335  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:  ufileu  22619  filufint  22620  uffixfr  22623  uffix2  22624  uffixsn  22625  hausflim  22681  fclsval  22708  isfcls  22709  fclsopn  22714  fclsfnflim  22727  filnetlem4  34119
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