Theorem ufilss 23792

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 6895	. . . 4 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝑋 ∈ dom UFil)
2		elpw2g 5288	. . . 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 23790	. . . 4 ⊢ (𝐹 ∈ (UFil‘𝑋) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋(𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹)))
5		eleq1 2816	. . . . . 6 ⊢ (𝑥 = 𝑆 → (𝑥 ∈ 𝐹 ↔ 𝑆 ∈ 𝐹))
6		difeq2 4083	. . . . . . 7 ⊢ (𝑥 = 𝑆 → (𝑋 ∖ 𝑥) = (𝑋 ∖ 𝑆))
7	6	eleq1d 2813	. . . . . 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 3585	. . . 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 3044   ∖ cdif 3911   ⊆ wss 3914  𝒫 cpw 4563  dom cdm 5638  ‘cfv 6511  Filcfil 23732  UFilcufil 23786

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 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fv 6519  df-ufil 23788

This theorem is referenced by:  ufilb  23793  trufil  23797  ufildr  23818