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Theorem ufilb 21459
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 21458 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆𝑋) → (𝑆𝐹 ∨ (𝑋𝑆) ∈ 𝐹))
21ord 390 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆𝑋) → (¬ 𝑆𝐹 → (𝑋𝑆) ∈ 𝐹))
3 ufilfil 21457 . . . 4 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
4 filfbas 21401 . . . 4 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
5 fbncp 21392 . . . . . 6 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑆𝐹) → ¬ (𝑋𝑆) ∈ 𝐹)
65ex 448 . . . . 5 (𝐹 ∈ (fBas‘𝑋) → (𝑆𝐹 → ¬ (𝑋𝑆) ∈ 𝐹))
76con2d 127 . . . 4 (𝐹 ∈ (fBas‘𝑋) → ((𝑋𝑆) ∈ 𝐹 → ¬ 𝑆𝐹))
83, 4, 73syl 18 . . 3 (𝐹 ∈ (UFil‘𝑋) → ((𝑋𝑆) ∈ 𝐹 → ¬ 𝑆𝐹))
98adantr 479 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆𝑋) → ((𝑋𝑆) ∈ 𝐹 → ¬ 𝑆𝐹))
102, 9impbid 200 1 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆𝑋) → (¬ 𝑆𝐹 ↔ (𝑋𝑆) ∈ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  wcel 1976  cdif 3533  wss 3536  cfv 5787  fBascfbas 19498  Filcfil 21398  UFilcufil 21452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-iota 5751  df-fun 5789  df-fv 5795  df-fbas 19507  df-fil 21399  df-ufil 21454
This theorem is referenced by:  ufilmax  21460  ufprim  21462  trufil  21463  ufileu  21472  cfinufil  21481  alexsublem  21597
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