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Theorem ufilfil 22447
Description: An ultrafilter is a filter. (Contributed by Jeff Hankins, 1-Dec-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)
Assertion
Ref Expression
ufilfil (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))

Proof of Theorem ufilfil
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 isufil 22446 . 2 (𝐹 ∈ (UFil‘𝑋) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋(𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹)))
21simplbi 498 1 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 843  wcel 2107  wral 3143  cdif 3937  𝒫 cpw 4542  cfv 6354  Filcfil 22388  UFilcufil 22442
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fv 6362  df-ufil 22444
This theorem is referenced by:  ufilb  22449  isufil2  22451  ufprim  22452  trufil  22453  ufileu  22462  filufint  22463  uffixfr  22466  uffix2  22467  uffixsn  22468  uffinfix  22470  cfinufil  22471  ufilen  22473  ufildr  22474  fmufil  22502  ufldom  22505  uffclsflim  22574  ufilcmp  22575  uffcfflf  22582  alexsublem  22587
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