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Theorem ufilb 17601
Description: The complement is in an ultrafilter iff the set is not. (Contributed by Mario Carneiro, 11-Dec-2013.) (Revised by Mario Carneiro, 29-Jul-2015.)
Assertion
Ref Expression
ufilb  |-  ( ( F  e.  ( UFil `  X )  /\  S  C_  X )  ->  ( -.  S  e.  F  <->  ( X  \  S )  e.  F ) )

Proof of Theorem ufilb
StepHypRef Expression
1 ufilss 17600 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  S  C_  X )  ->  ( S  e.  F  \/  ( X  \  S )  e.  F ) )
21ord 366 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  S  C_  X )  ->  ( -.  S  e.  F  ->  ( X  \  S
)  e.  F ) )
3 ufilfil 17599 . . . 4  |-  ( F  e.  ( UFil `  X
)  ->  F  e.  ( Fil `  X ) )
4 filfbas 17543 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  ( fBas `  X )
)
5 fbncp 17534 . . . . . 6  |-  ( ( F  e.  ( fBas `  X )  /\  S  e.  F )  ->  -.  ( X  \  S )  e.  F )
65ex 423 . . . . 5  |-  ( F  e.  ( fBas `  X
)  ->  ( S  e.  F  ->  -.  ( X  \  S )  e.  F ) )
76con2d 107 . . . 4  |-  ( F  e.  ( fBas `  X
)  ->  ( ( X  \  S )  e.  F  ->  -.  S  e.  F ) )
83, 4, 73syl 18 . . 3  |-  ( F  e.  ( UFil `  X
)  ->  ( ( X  \  S )  e.  F  ->  -.  S  e.  F ) )
98adantr 451 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  S  C_  X )  ->  (
( X  \  S
)  e.  F  ->  -.  S  e.  F
) )
102, 9impbid 183 1  |-  ( ( F  e.  ( UFil `  X )  /\  S  C_  X )  ->  ( -.  S  e.  F  <->  ( X  \  S )  e.  F ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684    \ cdif 3149    C_ wss 3152   ` cfv 5255   fBascfbas 17518   Filcfil 17540   UFilcufil 17594
This theorem is referenced by:  ufilmax  17602  ufprim  17604  trufil  17605  ufileu  17614  cfinufil  17623  alexsublem  17738
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-fbas 17520  df-fil 17541  df-ufil 17596
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