Theorem ufilss 23848

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) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐹 ∨ (𝑋 ∖ 𝑆) ∈ 𝐹))

Proof of Theorem ufilss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvdm 6918	. . . 4 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝑋 ∈ dom UFil)
2		elpw2g 5308	. . . 4 ⊢ (𝑋 ∈ dom UFil → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋))
3	1, 2	syl 17	. . 3 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋))
4		isufil 23846	. . . 4 ⊢ (𝐹 ∈ (UFil‘𝑋) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋(𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹)))
5		eleq1 2823	. . . . . 6 ⊢ (𝑥 = 𝑆 → (𝑥 ∈ 𝐹 ↔ 𝑆 ∈ 𝐹))
6		difeq2 4100	. . . . . . 7 ⊢ (𝑥 = 𝑆 → (𝑋 ∖ 𝑥) = (𝑋 ∖ 𝑆))
7	6	eleq1d 2820	. . . . . 6 ⊢ (𝑥 = 𝑆 → ((𝑋 ∖ 𝑥) ∈ 𝐹 ↔ (𝑋 ∖ 𝑆) ∈ 𝐹))
8	5, 7	orbi12d 918	. . . . 5 ⊢ (𝑥 = 𝑆 → ((𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹) ↔ (𝑆 ∈ 𝐹 ∨ (𝑋 ∖ 𝑆) ∈ 𝐹)))
9	8	rspccv 3603	. . . 4 ⊢ (∀𝑥 ∈ 𝒫 𝑋(𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹) → (𝑆 ∈ 𝒫 𝑋 → (𝑆 ∈ 𝐹 ∨ (𝑋 ∖ 𝑆) ∈ 𝐹)))
10	4, 9	simplbiim 504	. . 3 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑆 ∈ 𝒫 𝑋 → (𝑆 ∈ 𝐹 ∨ (𝑋 ∖ 𝑆) ∈ 𝐹)))
11	3, 10	sylbird 260	. 2 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑆 ⊆ 𝑋 → (𝑆 ∈ 𝐹 ∨ (𝑋 ∖ 𝑆) ∈ 𝐹)))
12	11	imp 406	1 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐹 ∨ (𝑋 ∖ 𝑆) ∈ 𝐹))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∀wral 3052   ∖ cdif 3928   ⊆ wss 3931  𝒫 cpw 4580  dom cdm 5659  ‘cfv 6536  Filcfil 23788  UFilcufil 23842

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fv 6544  df-ufil 23844

This theorem is referenced by:  ufilb  23849  trufil  23853  ufildr  23874