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Theorem ufilss 21519
 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 6130 . . . 4 (𝐹 ∈ (UFil‘𝑋) → 𝑋 ∈ dom UFil)
2 elpw2g 4754 . . . 4 (𝑋 ∈ dom UFil → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
31, 2syl 17 . . 3 (𝐹 ∈ (UFil‘𝑋) → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
4 isufil 21517 . . . . 5 (𝐹 ∈ (UFil‘𝑋) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋(𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹)))
54simprbi 479 . . . 4 (𝐹 ∈ (UFil‘𝑋) → ∀𝑥 ∈ 𝒫 𝑋(𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹))
6 eleq1 2676 . . . . . 6 (𝑥 = 𝑆 → (𝑥𝐹𝑆𝐹))
7 difeq2 3684 . . . . . . 7 (𝑥 = 𝑆 → (𝑋𝑥) = (𝑋𝑆))
87eleq1d 2672 . . . . . 6 (𝑥 = 𝑆 → ((𝑋𝑥) ∈ 𝐹 ↔ (𝑋𝑆) ∈ 𝐹))
96, 8orbi12d 742 . . . . 5 (𝑥 = 𝑆 → ((𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹) ↔ (𝑆𝐹 ∨ (𝑋𝑆) ∈ 𝐹)))
109rspccv 3279 . . . 4 (∀𝑥 ∈ 𝒫 𝑋(𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹) → (𝑆 ∈ 𝒫 𝑋 → (𝑆𝐹 ∨ (𝑋𝑆) ∈ 𝐹)))
115, 10syl 17 . . 3 (𝐹 ∈ (UFil‘𝑋) → (𝑆 ∈ 𝒫 𝑋 → (𝑆𝐹 ∨ (𝑋𝑆) ∈ 𝐹)))
123, 11sylbird 249 . 2 (𝐹 ∈ (UFil‘𝑋) → (𝑆𝑋 → (𝑆𝐹 ∨ (𝑋𝑆) ∈ 𝐹)))
1312imp 444 1 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆𝑋) → (𝑆𝐹 ∨ (𝑋𝑆) ∈ 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∖ cdif 3537   ⊆ wss 3540  𝒫 cpw 4108  dom cdm 5038  ‘cfv 5804  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-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-ufil 21515 This theorem is referenced by:  ufilb  21520  trufil  21524  ufildr  21545
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