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Theorem ufilfil 19595
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 19594 . 2  |-  ( F  e.  ( UFil `  X
)  <->  ( F  e.  ( Fil `  X
)  /\  A. x  e.  ~P  X ( x  e.  F  \/  ( X  \  x )  e.  F ) ) )
21simplbi 460 1  |-  ( F  e.  ( UFil `  X
)  ->  F  e.  ( Fil `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    e. wcel 1758   A.wral 2795    \ cdif 3425   ~Pcpw 3960   ` cfv 5518   Filcfil 19536   UFilcufil 19590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fv 5526  df-ufil 19592
This theorem is referenced by:  ufilb  19597  isufil2  19599  ufprim  19600  trufil  19601  ufileu  19610  filufint  19611  uffixfr  19614  uffix2  19615  uffixsn  19616  uffinfix  19618  cfinufil  19619  ufilen  19621  ufildr  19622  fmufil  19650  ufldom  19653  uffclsflim  19722  ufilcmp  19723  uffcfflf  19730  alexsublem  19734
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