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Theorem ufilss 23913
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 6943 . . . 4 (𝐹 ∈ (UFil‘𝑋) → 𝑋 ∈ dom UFil)
2 elpw2g 5333 . . . 4 (𝑋 ∈ dom UFil → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
31, 2syl 17 . . 3 (𝐹 ∈ (UFil‘𝑋) → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
4 isufil 23911 . . . 4 (𝐹 ∈ (UFil‘𝑋) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋(𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹)))
5 eleq1 2829 . . . . . 6 (𝑥 = 𝑆 → (𝑥𝐹𝑆𝐹))
6 difeq2 4120 . . . . . . 7 (𝑥 = 𝑆 → (𝑋𝑥) = (𝑋𝑆))
76eleq1d 2826 . . . . . 6 (𝑥 = 𝑆 → ((𝑋𝑥) ∈ 𝐹 ↔ (𝑋𝑆) ∈ 𝐹))
85, 7orbi12d 919 . . . . 5 (𝑥 = 𝑆 → ((𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹) ↔ (𝑆𝐹 ∨ (𝑋𝑆) ∈ 𝐹)))
98rspccv 3619 . . . 4 (∀𝑥 ∈ 𝒫 𝑋(𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹) → (𝑆 ∈ 𝒫 𝑋 → (𝑆𝐹 ∨ (𝑋𝑆) ∈ 𝐹)))
104, 9simplbiim 504 . . 3 (𝐹 ∈ (UFil‘𝑋) → (𝑆 ∈ 𝒫 𝑋 → (𝑆𝐹 ∨ (𝑋𝑆) ∈ 𝐹)))
113, 10sylbird 260 . 2 (𝐹 ∈ (UFil‘𝑋) → (𝑆𝑋 → (𝑆𝐹 ∨ (𝑋𝑆) ∈ 𝐹)))
1211imp 406 1 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆𝑋) → (𝑆𝐹 ∨ (𝑋𝑆) ∈ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  wral 3061  cdif 3948  wss 3951  𝒫 cpw 4600  dom cdm 5685  cfv 6561  Filcfil 23853  UFilcufil 23907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-ufil 23909
This theorem is referenced by:  ufilb  23914  trufil  23918  ufildr  23939
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