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Theorem ufilss 22441
Description: For any subset of the base set of an ultrafilter, either the set is in the ultrafilter or the complement is. (Contributed by Jeff Hankins, 1-Dec-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)
Assertion
Ref Expression
ufilss ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆𝑋) → (𝑆𝐹 ∨ (𝑋𝑆) ∈ 𝐹))

Proof of Theorem ufilss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elfvdm 6695 . . . 4 (𝐹 ∈ (UFil‘𝑋) → 𝑋 ∈ dom UFil)
2 elpw2g 5238 . . . 4 (𝑋 ∈ dom UFil → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
31, 2syl 17 . . 3 (𝐹 ∈ (UFil‘𝑋) → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
4 isufil 22439 . . . 4 (𝐹 ∈ (UFil‘𝑋) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋(𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹)))
5 eleq1 2897 . . . . . 6 (𝑥 = 𝑆 → (𝑥𝐹𝑆𝐹))
6 difeq2 4090 . . . . . . 7 (𝑥 = 𝑆 → (𝑋𝑥) = (𝑋𝑆))
76eleq1d 2894 . . . . . 6 (𝑥 = 𝑆 → ((𝑋𝑥) ∈ 𝐹 ↔ (𝑋𝑆) ∈ 𝐹))
85, 7orbi12d 912 . . . . 5 (𝑥 = 𝑆 → ((𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹) ↔ (𝑆𝐹 ∨ (𝑋𝑆) ∈ 𝐹)))
98rspccv 3617 . . . 4 (∀𝑥 ∈ 𝒫 𝑋(𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹) → (𝑆 ∈ 𝒫 𝑋 → (𝑆𝐹 ∨ (𝑋𝑆) ∈ 𝐹)))
104, 9simplbiim 505 . . 3 (𝐹 ∈ (UFil‘𝑋) → (𝑆 ∈ 𝒫 𝑋 → (𝑆𝐹 ∨ (𝑋𝑆) ∈ 𝐹)))
113, 10sylbird 261 . 2 (𝐹 ∈ (UFil‘𝑋) → (𝑆𝑋 → (𝑆𝐹 ∨ (𝑋𝑆) ∈ 𝐹)))
1211imp 407 1 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆𝑋) → (𝑆𝐹 ∨ (𝑋𝑆) ∈ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 841   = wceq 1528  wcel 2105  wral 3135  cdif 3930  wss 3933  𝒫 cpw 4535  dom cdm 5548  cfv 6348  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-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-ufil 22437
This theorem is referenced by:  ufilb  22442  trufil  22446  ufildr  22467
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