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Theorem ufilss 21619
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 6177 . . . 4 (𝐹 ∈ (UFil‘𝑋) → 𝑋 ∈ dom UFil)
2 elpw2g 4787 . . . 4 (𝑋 ∈ dom UFil → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
31, 2syl 17 . . 3 (𝐹 ∈ (UFil‘𝑋) → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
4 isufil 21617 . . . . 5 (𝐹 ∈ (UFil‘𝑋) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋(𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹)))
54simprbi 480 . . . 4 (𝐹 ∈ (UFil‘𝑋) → ∀𝑥 ∈ 𝒫 𝑋(𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹))
6 eleq1 2686 . . . . . 6 (𝑥 = 𝑆 → (𝑥𝐹𝑆𝐹))
7 difeq2 3700 . . . . . . 7 (𝑥 = 𝑆 → (𝑋𝑥) = (𝑋𝑆))
87eleq1d 2683 . . . . . 6 (𝑥 = 𝑆 → ((𝑋𝑥) ∈ 𝐹 ↔ (𝑋𝑆) ∈ 𝐹))
96, 8orbi12d 745 . . . . 5 (𝑥 = 𝑆 → ((𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹) ↔ (𝑆𝐹 ∨ (𝑋𝑆) ∈ 𝐹)))
109rspccv 3292 . . . 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 250 . 2 (𝐹 ∈ (UFil‘𝑋) → (𝑆𝑋 → (𝑆𝐹 ∨ (𝑋𝑆) ∈ 𝐹)))
1312imp 445 1 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆𝑋) → (𝑆𝐹 ∨ (𝑋𝑆) ∈ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wcel 1987  wral 2907  cdif 3552  wss 3555  𝒫 cpw 4130  dom cdm 5074  cfv 5847  Filcfil 21559  UFilcufil 21613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fv 5855  df-ufil 21615
This theorem is referenced by:  ufilb  21620  trufil  21624  ufildr  21645
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