Theorem ufilb 23929

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 23928	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐹 ∨ (𝑋 ∖ 𝑆) ∈ 𝐹))
2	1	ord 864	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (¬ 𝑆 ∈ 𝐹 → (𝑋 ∖ 𝑆) ∈ 𝐹))
3		ufilfil 23927	. . . 4 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
4		filfbas 23871	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
5		fbncp 23862	. . . . . 6 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑆 ∈ 𝐹) → ¬ (𝑋 ∖ 𝑆) ∈ 𝐹)
6	5	ex 412	. . . . 5 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑆 ∈ 𝐹 → ¬ (𝑋 ∖ 𝑆) ∈ 𝐹))
7	6	con2d 134	. . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → ((𝑋 ∖ 𝑆) ∈ 𝐹 → ¬ 𝑆 ∈ 𝐹))
8	3, 4, 7	3syl 18	. . 3 ⊢ (𝐹 ∈ (UFil‘𝑋) → ((𝑋 ∖ 𝑆) ∈ 𝐹 → ¬ 𝑆 ∈ 𝐹))
9	8	adantr 480	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆 ⊆ 𝑋) → ((𝑋 ∖ 𝑆) ∈ 𝐹 → ¬ 𝑆 ∈ 𝐹))
10	2, 9	impbid 212	1 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (¬ 𝑆 ∈ 𝐹 ↔ (𝑋 ∖ 𝑆) ∈ 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2105   ∖ cdif 3959   ⊆ wss 3962  ‘cfv 6562  fBascfbas 21369  Filcfil 23868  UFilcufil 23922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fv 6570  df-fbas 21378  df-fil 23869  df-ufil 23924

This theorem is referenced by:  ufilmax  23930  ufprim  23932  trufil  23933  ufileu  23942  cfinufil  23951  alexsublem  24067