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Theorem ufilfil 20974
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 20973 . 2  |-  ( F  e.  ( UFil `  X
)  <->  ( F  e.  ( Fil `  X
)  /\  A. x  e.  ~P  X ( x  e.  F  \/  ( X  \  x )  e.  F ) ) )
21simplbi 466 1  |-  ( F  e.  ( UFil `  X
)  ->  F  e.  ( Fil `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 374    e. wcel 1898   A.wral 2749    \ cdif 3413   ~Pcpw 3963   ` cfv 5605   Filcfil 20915   UFilcufil 20969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-iota 5569  df-fun 5607  df-fv 5613  df-ufil 20971
This theorem is referenced by:  ufilb  20976  isufil2  20978  ufprim  20979  trufil  20980  ufileu  20989  filufint  20990  uffixfr  20993  uffix2  20994  uffixsn  20995  uffinfix  20997  cfinufil  20998  ufilen  21000  ufildr  21001  fmufil  21029  ufldom  21032  uffclsflim  21101  ufilcmp  21102  uffcfflf  21109  alexsublem  21114
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