MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ufilb Structured version   Visualization version   GIF version

Theorem ufilb 22442
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 22441 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆𝑋) → (𝑆𝐹 ∨ (𝑋𝑆) ∈ 𝐹))
21ord 858 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆𝑋) → (¬ 𝑆𝐹 → (𝑋𝑆) ∈ 𝐹))
3 ufilfil 22440 . . . 4 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
4 filfbas 22384 . . . 4 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
5 fbncp 22375 . . . . . 6 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑆𝐹) → ¬ (𝑋𝑆) ∈ 𝐹)
65ex 413 . . . . 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 481 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆𝑋) → ((𝑋𝑆) ∈ 𝐹 → ¬ 𝑆𝐹))
102, 9impbid 213 1 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆𝑋) → (¬ 𝑆𝐹 ↔ (𝑋𝑆) ∈ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wcel 2105  cdif 3930  wss 3933  cfv 6348  fBascfbas 20461  Filcfil 22381  UFilcufil 22435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  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 20470  df-fil 22382  df-ufil 22437
This theorem is referenced by:  ufilmax  22443  ufprim  22445  trufil  22446  ufileu  22455  cfinufil  22464  alexsublem  22580
  Copyright terms: Public domain W3C validator