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Theorem ufilb 21520
 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 21519 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆𝑋) → (𝑆𝐹 ∨ (𝑋𝑆) ∈ 𝐹))
21ord 391 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆𝑋) → (¬ 𝑆𝐹 → (𝑋𝑆) ∈ 𝐹))
3 ufilfil 21518 . . . 4 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
4 filfbas 21462 . . . 4 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
5 fbncp 21453 . . . . . 6 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑆𝐹) → ¬ (𝑋𝑆) ∈ 𝐹)
65ex 449 . . . . 5 (𝐹 ∈ (fBas‘𝑋) → (𝑆𝐹 → ¬ (𝑋𝑆) ∈ 𝐹))
76con2d 128 . . . 4 (𝐹 ∈ (fBas‘𝑋) → ((𝑋𝑆) ∈ 𝐹 → ¬ 𝑆𝐹))
83, 4, 73syl 18 . . 3 (𝐹 ∈ (UFil‘𝑋) → ((𝑋𝑆) ∈ 𝐹 → ¬ 𝑆𝐹))
98adantr 480 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆𝑋) → ((𝑋𝑆) ∈ 𝐹 → ¬ 𝑆𝐹))
102, 9impbid 201 1 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆𝑋) → (¬ 𝑆𝐹 ↔ (𝑋𝑆) ∈ 𝐹))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977   ∖ cdif 3537   ⊆ wss 3540  ‘cfv 5804  fBascfbas 19555  Filcfil 21459  UFilcufil 21513 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-fbas 19564  df-fil 21460  df-ufil 21515 This theorem is referenced by:  ufilmax  21521  ufprim  21523  trufil  21524  ufileu  21533  cfinufil  21542  alexsublem  21658
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