Theorem ufprim 23933

Description: An ultrafilter is a prime filter. (Contributed by Jeff Hankins, 1-Jan-2010.) (Revised by Mario Carneiro, 2-Aug-2015.)

Assertion

Ref	Expression
ufprim	⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((𝐴 ∈ 𝐹 ∨ 𝐵 ∈ 𝐹) ↔ (𝐴 ∪ 𝐵) ∈ 𝐹))

Proof of Theorem ufprim

Step	Hyp	Ref	Expression
1		ufilfil 23928	. . . . . . 7 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
2	1	3ad2ant1 1132	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → 𝐹 ∈ (Fil‘𝑋))
3	2	adantr 480	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
4		simpr 484	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ∈ 𝐹)
5		unss 4200	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ↔ (𝐴 ∪ 𝐵) ⊆ 𝑋)
6	5	biimpi 216	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐴 ∪ 𝐵) ⊆ 𝑋)
7	6	3adant1 1129	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐴 ∪ 𝐵) ⊆ 𝑋)
8	7	adantr 480	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐴 ∪ 𝐵) ⊆ 𝑋)
9		ssun1 4188	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
10	9	a1i 11	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ (𝐴 ∪ 𝐵))
11		filss 23877	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∈ 𝐹 ∧ (𝐴 ∪ 𝐵) ⊆ 𝑋 ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵))) → (𝐴 ∪ 𝐵) ∈ 𝐹)
12	3, 4, 8, 10, 11	syl13anc 1371	. . . 4 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐴 ∪ 𝐵) ∈ 𝐹)
13	12	ex 412	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐴 ∈ 𝐹 → (𝐴 ∪ 𝐵) ∈ 𝐹))
14	2	adantr 480	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐵 ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
15		simpr 484	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐵 ∈ 𝐹) → 𝐵 ∈ 𝐹)
16	7	adantr 480	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐵 ∈ 𝐹) → (𝐴 ∪ 𝐵) ⊆ 𝑋)
17		ssun2 4189	. . . . . 6 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
18	17	a1i 11	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐵 ∈ 𝐹) → 𝐵 ⊆ (𝐴 ∪ 𝐵))
19		filss 23877	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐵 ∈ 𝐹 ∧ (𝐴 ∪ 𝐵) ⊆ 𝑋 ∧ 𝐵 ⊆ (𝐴 ∪ 𝐵))) → (𝐴 ∪ 𝐵) ∈ 𝐹)
20	14, 15, 16, 18, 19	syl13anc 1371	. . . 4 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐵 ∈ 𝐹) → (𝐴 ∪ 𝐵) ∈ 𝐹)
21	20	ex 412	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐵 ∈ 𝐹 → (𝐴 ∪ 𝐵) ∈ 𝐹))
22	13, 21	jaod 859	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((𝐴 ∈ 𝐹 ∨ 𝐵 ∈ 𝐹) → (𝐴 ∪ 𝐵) ∈ 𝐹))
23		ufilb 23930	. . . . . . 7 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (¬ 𝐴 ∈ 𝐹 ↔ (𝑋 ∖ 𝐴) ∈ 𝐹))
24	23	3adant3 1131	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (¬ 𝐴 ∈ 𝐹 ↔ (𝑋 ∖ 𝐴) ∈ 𝐹))
25	24	adantr 480	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹) → (¬ 𝐴 ∈ 𝐹 ↔ (𝑋 ∖ 𝐴) ∈ 𝐹))
26	2	3ad2ant1 1132	. . . . . . 7 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹 ∧ (𝑋 ∖ 𝐴) ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
27		difun2 4487	. . . . . . . . . . 11 ⊢ ((𝐵 ∪ 𝐴) ∖ 𝐴) = (𝐵 ∖ 𝐴)
28		uncom 4168	. . . . . . . . . . . 12 ⊢ (𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵)
29	28	difeq1i 4132	. . . . . . . . . . 11 ⊢ ((𝐵 ∪ 𝐴) ∖ 𝐴) = ((𝐴 ∪ 𝐵) ∖ 𝐴)
30	27, 29	eqtr3i 2765	. . . . . . . . . 10 ⊢ (𝐵 ∖ 𝐴) = ((𝐴 ∪ 𝐵) ∖ 𝐴)
31	30	ineq2i 4225	. . . . . . . . 9 ⊢ (𝑋 ∩ (𝐵 ∖ 𝐴)) = (𝑋 ∩ ((𝐴 ∪ 𝐵) ∖ 𝐴))
32		indifcom 4289	. . . . . . . . 9 ⊢ (𝐵 ∩ (𝑋 ∖ 𝐴)) = (𝑋 ∩ (𝐵 ∖ 𝐴))
33		indifcom 4289	. . . . . . . . 9 ⊢ ((𝐴 ∪ 𝐵) ∩ (𝑋 ∖ 𝐴)) = (𝑋 ∩ ((𝐴 ∪ 𝐵) ∖ 𝐴))
34	31, 32, 33	3eqtr4i 2773	. . . . . . . 8 ⊢ (𝐵 ∩ (𝑋 ∖ 𝐴)) = ((𝐴 ∪ 𝐵) ∩ (𝑋 ∖ 𝐴))
35		filin 23878	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹 ∧ (𝑋 ∖ 𝐴) ∈ 𝐹) → ((𝐴 ∪ 𝐵) ∩ (𝑋 ∖ 𝐴)) ∈ 𝐹)
36	2, 35	syl3an1 1162	. . . . . . . 8 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹 ∧ (𝑋 ∖ 𝐴) ∈ 𝐹) → ((𝐴 ∪ 𝐵) ∩ (𝑋 ∖ 𝐴)) ∈ 𝐹)
37	34, 36	eqeltrid 2843	. . . . . . 7 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹 ∧ (𝑋 ∖ 𝐴) ∈ 𝐹) → (𝐵 ∩ (𝑋 ∖ 𝐴)) ∈ 𝐹)
38		simp13 1204	. . . . . . 7 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹 ∧ (𝑋 ∖ 𝐴) ∈ 𝐹) → 𝐵 ⊆ 𝑋)
39		inss1 4245	. . . . . . . 8 ⊢ (𝐵 ∩ (𝑋 ∖ 𝐴)) ⊆ 𝐵
40	39	a1i 11	. . . . . . 7 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹 ∧ (𝑋 ∖ 𝐴) ∈ 𝐹) → (𝐵 ∩ (𝑋 ∖ 𝐴)) ⊆ 𝐵)
41		filss 23877	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ((𝐵 ∩ (𝑋 ∖ 𝐴)) ∈ 𝐹 ∧ 𝐵 ⊆ 𝑋 ∧ (𝐵 ∩ (𝑋 ∖ 𝐴)) ⊆ 𝐵)) → 𝐵 ∈ 𝐹)
42	26, 37, 38, 40, 41	syl13anc 1371	. . . . . 6 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹 ∧ (𝑋 ∖ 𝐴) ∈ 𝐹) → 𝐵 ∈ 𝐹)
43	42	3expia 1120	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹) → ((𝑋 ∖ 𝐴) ∈ 𝐹 → 𝐵 ∈ 𝐹))
44	25, 43	sylbid 240	. . . 4 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹) → (¬ 𝐴 ∈ 𝐹 → 𝐵 ∈ 𝐹))
45	44	orrd 863	. . 3 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹) → (𝐴 ∈ 𝐹 ∨ 𝐵 ∈ 𝐹))
46	45	ex 412	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((𝐴 ∪ 𝐵) ∈ 𝐹 → (𝐴 ∈ 𝐹 ∨ 𝐵 ∈ 𝐹)))
47	22, 46	impbid 212	1 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((𝐴 ∈ 𝐹 ∨ 𝐵 ∈ 𝐹) ↔ (𝐴 ∪ 𝐵) ∈ 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   ∈ wcel 2106   ∖ cdif 3960   ∪ cun 3961   ∩ cin 3962   ⊆ wss 3963  ‘cfv 6563  Filcfil 23869  UFilcufil 23923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-fbas 21379  df-fil 23870  df-ufil 23925

This theorem is referenced by: (None)