Theorem ufprim 22760

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 22755	. . . . . . 7 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
2	1	3ad2ant1 1135	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → 𝐹 ∈ (Fil‘𝑋))
3	2	adantr 484	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
4		simpr 488	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ∈ 𝐹)
5		unss 4084	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ↔ (𝐴 ∪ 𝐵) ⊆ 𝑋)
6	5	biimpi 219	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐴 ∪ 𝐵) ⊆ 𝑋)
7	6	3adant1 1132	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐴 ∪ 𝐵) ⊆ 𝑋)
8	7	adantr 484	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐴 ∪ 𝐵) ⊆ 𝑋)
9		ssun1 4072	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
10	9	a1i 11	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ (𝐴 ∪ 𝐵))
11		filss 22704	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∈ 𝐹 ∧ (𝐴 ∪ 𝐵) ⊆ 𝑋 ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵))) → (𝐴 ∪ 𝐵) ∈ 𝐹)
12	3, 4, 8, 10, 11	syl13anc 1374	. . . 4 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐴 ∪ 𝐵) ∈ 𝐹)
13	12	ex 416	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐴 ∈ 𝐹 → (𝐴 ∪ 𝐵) ∈ 𝐹))
14	2	adantr 484	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐵 ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
15		simpr 488	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐵 ∈ 𝐹) → 𝐵 ∈ 𝐹)
16	7	adantr 484	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐵 ∈ 𝐹) → (𝐴 ∪ 𝐵) ⊆ 𝑋)
17		ssun2 4073	. . . . . 6 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
18	17	a1i 11	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐵 ∈ 𝐹) → 𝐵 ⊆ (𝐴 ∪ 𝐵))
19		filss 22704	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐵 ∈ 𝐹 ∧ (𝐴 ∪ 𝐵) ⊆ 𝑋 ∧ 𝐵 ⊆ (𝐴 ∪ 𝐵))) → (𝐴 ∪ 𝐵) ∈ 𝐹)
20	14, 15, 16, 18, 19	syl13anc 1374	. . . 4 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐵 ∈ 𝐹) → (𝐴 ∪ 𝐵) ∈ 𝐹)
21	20	ex 416	. . 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 22757	. . . . . . 7 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (¬ 𝐴 ∈ 𝐹 ↔ (𝑋 ∖ 𝐴) ∈ 𝐹))
24	23	3adant3 1134	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (¬ 𝐴 ∈ 𝐹 ↔ (𝑋 ∖ 𝐴) ∈ 𝐹))
25	24	adantr 484	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹) → (¬ 𝐴 ∈ 𝐹 ↔ (𝑋 ∖ 𝐴) ∈ 𝐹))
26	2	3ad2ant1 1135	. . . . . . 7 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹 ∧ (𝑋 ∖ 𝐴) ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
27		difun2 4381	. . . . . . . . . . 11 ⊢ ((𝐵 ∪ 𝐴) ∖ 𝐴) = (𝐵 ∖ 𝐴)
28		uncom 4053	. . . . . . . . . . . 12 ⊢ (𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵)
29	28	difeq1i 4019	. . . . . . . . . . 11 ⊢ ((𝐵 ∪ 𝐴) ∖ 𝐴) = ((𝐴 ∪ 𝐵) ∖ 𝐴)
30	27, 29	eqtr3i 2761	. . . . . . . . . 10 ⊢ (𝐵 ∖ 𝐴) = ((𝐴 ∪ 𝐵) ∖ 𝐴)
31	30	ineq2i 4110	. . . . . . . . 9 ⊢ (𝑋 ∩ (𝐵 ∖ 𝐴)) = (𝑋 ∩ ((𝐴 ∪ 𝐵) ∖ 𝐴))
32		indifcom 4173	. . . . . . . . 9 ⊢ (𝐵 ∩ (𝑋 ∖ 𝐴)) = (𝑋 ∩ (𝐵 ∖ 𝐴))
33		indifcom 4173	. . . . . . . . 9 ⊢ ((𝐴 ∪ 𝐵) ∩ (𝑋 ∖ 𝐴)) = (𝑋 ∩ ((𝐴 ∪ 𝐵) ∖ 𝐴))
34	31, 32, 33	3eqtr4i 2769	. . . . . . . 8 ⊢ (𝐵 ∩ (𝑋 ∖ 𝐴)) = ((𝐴 ∪ 𝐵) ∩ (𝑋 ∖ 𝐴))
35		filin 22705	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹 ∧ (𝑋 ∖ 𝐴) ∈ 𝐹) → ((𝐴 ∪ 𝐵) ∩ (𝑋 ∖ 𝐴)) ∈ 𝐹)
36	2, 35	syl3an1 1165	. . . . . . . 8 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹 ∧ (𝑋 ∖ 𝐴) ∈ 𝐹) → ((𝐴 ∪ 𝐵) ∩ (𝑋 ∖ 𝐴)) ∈ 𝐹)
37	34, 36	eqeltrid 2835	. . . . . . 7 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹 ∧ (𝑋 ∖ 𝐴) ∈ 𝐹) → (𝐵 ∩ (𝑋 ∖ 𝐴)) ∈ 𝐹)
38		simp13 1207	. . . . . . 7 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹 ∧ (𝑋 ∖ 𝐴) ∈ 𝐹) → 𝐵 ⊆ 𝑋)
39		inss1 4129	. . . . . . . 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 22704	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ((𝐵 ∩ (𝑋 ∖ 𝐴)) ∈ 𝐹 ∧ 𝐵 ⊆ 𝑋 ∧ (𝐵 ∩ (𝑋 ∖ 𝐴)) ⊆ 𝐵)) → 𝐵 ∈ 𝐹)
42	26, 37, 38, 40, 41	syl13anc 1374	. . . . . 6 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹 ∧ (𝑋 ∖ 𝐴) ∈ 𝐹) → 𝐵 ∈ 𝐹)
43	42	3expia 1123	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹) → ((𝑋 ∖ 𝐴) ∈ 𝐹 → 𝐵 ∈ 𝐹))
44	25, 43	sylbid 243	. . . 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 416	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((𝐴 ∪ 𝐵) ∈ 𝐹 → (𝐴 ∈ 𝐹 ∨ 𝐵 ∈ 𝐹)))
47	22, 46	impbid 215	1 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((𝐴 ∈ 𝐹 ∨ 𝐵 ∈ 𝐹) ↔ (𝐴 ∪ 𝐵) ∈ 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 847   ∧ w3a 1089   ∈ wcel 2112   ∖ cdif 3850   ∪ cun 3851   ∩ cin 3852   ⊆ wss 3853  ‘cfv 6358  Filcfil 22696  UFilcufil 22750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-nel 3037  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fv 6366  df-fbas 20314  df-fil 22697  df-ufil 22752

This theorem is referenced by: (None)