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Theorem ufilfil 20490
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  e.  ( UFil `  X
)  ->  F  e.  ( Fil `  X ) )

Proof of Theorem ufilfil
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 isufil 20489 . 2  |-  ( F  e.  ( UFil `  X
)  <->  ( F  e.  ( Fil `  X
)  /\  A. x  e.  ~P  X ( x  e.  F  \/  ( X  \  x )  e.  F ) ) )
21simplbi 458 1  |-  ( F  e.  ( UFil `  X
)  ->  F  e.  ( Fil `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    e. wcel 1826   A.wral 2732    \ cdif 3386   ~Pcpw 3927   ` cfv 5496   Filcfil 20431   UFilcufil 20485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fv 5504  df-ufil 20487
This theorem is referenced by:  ufilb  20492  isufil2  20494  ufprim  20495  trufil  20496  ufileu  20505  filufint  20506  uffixfr  20509  uffix2  20510  uffixsn  20511  uffinfix  20513  cfinufil  20514  ufilen  20516  ufildr  20517  fmufil  20545  ufldom  20548  uffclsflim  20617  ufilcmp  20618  uffcfflf  20625  alexsublem  20629
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