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Theorem ufilfil 17858
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 17857 . 2  |-  ( F  e.  ( UFil `  X
)  <->  ( F  e.  ( Fil `  X
)  /\  A. x  e.  ~P  X ( x  e.  F  \/  ( X  \  x )  e.  F ) ) )
21simplbi 447 1  |-  ( F  e.  ( UFil `  X
)  ->  F  e.  ( Fil `  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    e. wcel 1717   A.wral 2650    \ cdif 3261   ~Pcpw 3743   ` cfv 5395   Filcfil 17799   UFilcufil 17853
This theorem is referenced by:  ufilb  17860  isufil2  17862  ufprim  17863  trufil  17864  ufileu  17873  filufint  17874  uffixfr  17877  uffix2  17878  uffixsn  17879  uffinfix  17881  cfinufil  17882  ufilen  17884  ufildr  17885  fmufil  17913  ufldom  17916  uffclsflim  17985  ufilcmp  17986  uffcfflf  17993  alexsublem  17997
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fv 5403  df-ufil 17855
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