Theorem ufilss 24041

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 6915	. . . 4 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝑋 ∈ dom UFil)
2		elpw2g 5303	. . . 4 ⊢ (𝑋 ∈ dom UFil → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋))
3	1, 2	syl 18	. . 3 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋))
4		isufil 24039	. . . 4 ⊢ (𝐹 ∈ (UFil‘𝑋) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋(𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹)))
5		eleq1 2849	. . . . . 6 ⊢ (𝑥 = 𝑆 → (𝑥 ∈ 𝐹 ↔ 𝑆 ∈ 𝐹))
6		difeq2 4074	. . . . . . 7 ⊢ (𝑥 = 𝑆 → (𝑋 ∖ 𝑥) = (𝑋 ∖ 𝑆))
7	6	eleq1d 2846	. . . . . 6 ⊢ (𝑥 = 𝑆 → ((𝑋 ∖ 𝑥) ∈ 𝐹 ↔ (𝑋 ∖ 𝑆) ∈ 𝐹))
8	5, 7	orbi12d 931	. . . . 5 ⊢ (𝑥 = 𝑆 → ((𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹) ↔ (𝑆 ∈ 𝐹 ∨ (𝑋 ∖ 𝑆) ∈ 𝐹)))
9	8	rspccv 3577	. . . 4 ⊢ (∀𝑥 ∈ 𝒫 𝑋(𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹) → (𝑆 ∈ 𝒫 𝑋 → (𝑆 ∈ 𝐹 ∨ (𝑋 ∖ 𝑆) ∈ 𝐹)))
10	4, 9	simplbiim 513	. . 3 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑆 ∈ 𝒫 𝑋 → (𝑆 ∈ 𝐹 ∨ (𝑋 ∖ 𝑆) ∈ 𝐹)))
11	3, 10	sylbird 263	. 2 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑆 ⊆ 𝑋 → (𝑆 ∈ 𝐹 ∨ (𝑋 ∖ 𝑆) ∈ 𝐹)))
12	11	imp 411	1 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐹 ∨ (𝑋 ∖ 𝑆) ∈ 𝐹))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1568   ∈ wcel 2141  ∀wral 3077   ∖ cdif 3901   ⊆ wss 3904  𝒫 cpw 4561  dom cdm 5661  ‘cfv 6536  Filcfil 23981  UFilcufil 24035

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fv 6544  df-ufil 24037

This theorem is referenced by:  ufilb  24042  trufil  24046  ufildr  24067