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Theorem ufilb 22506
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 ∈ (UFil‘𝑋) ∧ 𝑆𝑋) → (¬ 𝑆𝐹 ↔ (𝑋𝑆) ∈ 𝐹))

Proof of Theorem ufilb
StepHypRef Expression
1 ufilss 22505 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆𝑋) → (𝑆𝐹 ∨ (𝑋𝑆) ∈ 𝐹))
21ord 860 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆𝑋) → (¬ 𝑆𝐹 → (𝑋𝑆) ∈ 𝐹))
3 ufilfil 22504 . . . 4 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
4 filfbas 22448 . . . 4 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
5 fbncp 22439 . . . . . 6 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑆𝐹) → ¬ (𝑋𝑆) ∈ 𝐹)
65ex 415 . . . . 5 (𝐹 ∈ (fBas‘𝑋) → (𝑆𝐹 → ¬ (𝑋𝑆) ∈ 𝐹))
76con2d 136 . . . 4 (𝐹 ∈ (fBas‘𝑋) → ((𝑋𝑆) ∈ 𝐹 → ¬ 𝑆𝐹))
83, 4, 73syl 18 . . 3 (𝐹 ∈ (UFil‘𝑋) → ((𝑋𝑆) ∈ 𝐹 → ¬ 𝑆𝐹))
98adantr 483 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆𝑋) → ((𝑋𝑆) ∈ 𝐹 → ¬ 𝑆𝐹))
102, 9impbid 214 1 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆𝑋) → (¬ 𝑆𝐹 ↔ (𝑋𝑆) ∈ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wcel 2107  cdif 3931  wss 3934  cfv 6348  fBascfbas 20525  Filcfil 22445  UFilcufil 22499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fv 6356  df-fbas 20534  df-fil 22446  df-ufil 22501
This theorem is referenced by:  ufilmax  22507  ufprim  22509  trufil  22510  ufileu  22519  cfinufil  22528  alexsublem  22644
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