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Theorem ufilss 22513
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 6681 . . . 4 (𝐹 ∈ (UFil‘𝑋) → 𝑋 ∈ dom UFil)
2 elpw2g 5214 . . . 4 (𝑋 ∈ dom UFil → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
31, 2syl 17 . . 3 (𝐹 ∈ (UFil‘𝑋) → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
4 isufil 22511 . . . 4 (𝐹 ∈ (UFil‘𝑋) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋(𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹)))
5 eleq1 2880 . . . . . 6 (𝑥 = 𝑆 → (𝑥𝐹𝑆𝐹))
6 difeq2 4047 . . . . . . 7 (𝑥 = 𝑆 → (𝑋𝑥) = (𝑋𝑆))
76eleq1d 2877 . . . . . 6 (𝑥 = 𝑆 → ((𝑋𝑥) ∈ 𝐹 ↔ (𝑋𝑆) ∈ 𝐹))
85, 7orbi12d 916 . . . . 5 (𝑥 = 𝑆 → ((𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹) ↔ (𝑆𝐹 ∨ (𝑋𝑆) ∈ 𝐹)))
98rspccv 3571 . . . 4 (∀𝑥 ∈ 𝒫 𝑋(𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹) → (𝑆 ∈ 𝒫 𝑋 → (𝑆𝐹 ∨ (𝑋𝑆) ∈ 𝐹)))
104, 9simplbiim 508 . . 3 (𝐹 ∈ (UFil‘𝑋) → (𝑆 ∈ 𝒫 𝑋 → (𝑆𝐹 ∨ (𝑋𝑆) ∈ 𝐹)))
113, 10sylbird 263 . 2 (𝐹 ∈ (UFil‘𝑋) → (𝑆𝑋 → (𝑆𝐹 ∨ (𝑋𝑆) ∈ 𝐹)))
1211imp 410 1 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆𝑋) → (𝑆𝐹 ∨ (𝑋𝑆) ∈ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wcel 2112  wral 3109  cdif 3881  wss 3884  𝒫 cpw 4500  dom cdm 5523  cfv 6328  Filcfil 22453  UFilcufil 22507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fv 6336  df-ufil 22509
This theorem is referenced by:  ufilb  22514  trufil  22518  ufildr  22539
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