Theorem ufilb 24074

Description: The complement is in an ultrafilter iff the set is not. (Contributed by Mario Carneiro, 11-Dec-2013.) (Revised by Mario Carneiro, 29-Jul-2015.)

Assertion

Ref	Expression
ufilb	⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (¬ 𝑆 ∈ 𝐹 ↔ (𝑋 ∖ 𝑆) ∈ 𝐹))

Proof of Theorem ufilb

Step	Hyp	Ref	Expression
1		ufilss 24073	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐹 ∨ (𝑋 ∖ 𝑆) ∈ 𝐹))
2	1	ord 877	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (¬ 𝑆 ∈ 𝐹 → (𝑋 ∖ 𝑆) ∈ 𝐹))
3		ufilfil 24072	. . . 4 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
4		filfbas 24016	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
5		fbncp 24007	. . . . . 6 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑆 ∈ 𝐹) → ¬ (𝑋 ∖ 𝑆) ∈ 𝐹)
6	5	ex 417	. . . . 5 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑆 ∈ 𝐹 → ¬ (𝑋 ∖ 𝑆) ∈ 𝐹))
7	6	con2d 135	. . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → ((𝑋 ∖ 𝑆) ∈ 𝐹 → ¬ 𝑆 ∈ 𝐹))
8	3, 4, 7	3syl 19	. . 3 ⊢ (𝐹 ∈ (UFil‘𝑋) → ((𝑋 ∖ 𝑆) ∈ 𝐹 → ¬ 𝑆 ∈ 𝐹))
9	8	adantr 485	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆 ⊆ 𝑋) → ((𝑋 ∖ 𝑆) ∈ 𝐹 → ¬ 𝑆 ∈ 𝐹))
10	2, 9	impbid 215	1 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (¬ 𝑆 ∈ 𝐹 ↔ (𝑋 ∖ 𝑆) ∈ 𝐹))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2142   ∖ cdif 3901   ⊆ wss 3904  ‘cfv 6536  fBascfbas 21521  Filcfil 24013  UFilcufil 24067

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  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 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fv 6544  df-fbas 21530  df-fil 24014  df-ufil 24069

This theorem is used by:  ufilmax  24075  ufprim  24077  trufil  24078  ufileu  24087  cfinufil  24096  alexsublem  24212