Theorem ufilb 24226

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 24225	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐹 ∨ (𝑋 ∖ 𝑆) ∈ 𝐹))
2	1	ord 878	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (¬ 𝑆 ∈ 𝐹 → (𝑋 ∖ 𝑆) ∈ 𝐹))
3		ufilfil 24224	. . . 4 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
4		filfbas 24168	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
5		fbncp 24159	. . . . . 6 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑆 ∈ 𝐹) → ¬ (𝑋 ∖ 𝑆) ∈ 𝐹)
6	5	ex 418	. . . . 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 486	. 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 401   ∈ wcel 2144   ⊆ wss 3895   ∖ cdif 4045  ‘cfv 6538  fBascfbas 21667  Filcfil 24165  UFilcufil 24219

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fv 6546  df-fbas 21676  df-fil 24166  df-ufil 24221

This theorem is used by:  ufilmax  24227  ufprim  24229  trufil  24230  ufileu  24239  cfinufil  24248  alexsublem  24364