Theorem ufilb 24024

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 24023	. . 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 24022	. . . 4 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
4		filfbas 23966	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
5		fbncp 23957	. . . . . 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))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2145   ∖ cdif 3904   ⊆ wss 3907  ‘cfv 6525  fBascfbas 21470  Filcfil 23963  UFilcufil 24017

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fv 6533  df-fbas 21479  df-fil 23964  df-ufil 24019

This theorem is referenced by:  ufilmax  24025  ufprim  24027  trufil  24028  ufileu  24037  cfinufil  24046  alexsublem  24162