Theorem ufprim 22517

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 22512	. . . . . . 7 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
2	1	3ad2ant1 1129	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → 𝐹 ∈ (Fil‘𝑋))
3	2	adantr 483	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
4		simpr 487	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ∈ 𝐹)
5		unss 4160	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ↔ (𝐴 ∪ 𝐵) ⊆ 𝑋)
6	5	biimpi 218	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐴 ∪ 𝐵) ⊆ 𝑋)
7	6	3adant1 1126	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐴 ∪ 𝐵) ⊆ 𝑋)
8	7	adantr 483	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐴 ∪ 𝐵) ⊆ 𝑋)
9		ssun1 4148	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
10	9	a1i 11	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ (𝐴 ∪ 𝐵))
11		filss 22461	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∈ 𝐹 ∧ (𝐴 ∪ 𝐵) ⊆ 𝑋 ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵))) → (𝐴 ∪ 𝐵) ∈ 𝐹)
12	3, 4, 8, 10, 11	syl13anc 1368	. . . 4 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐴 ∪ 𝐵) ∈ 𝐹)
13	12	ex 415	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐴 ∈ 𝐹 → (𝐴 ∪ 𝐵) ∈ 𝐹))
14	2	adantr 483	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐵 ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
15		simpr 487	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐵 ∈ 𝐹) → 𝐵 ∈ 𝐹)
16	7	adantr 483	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐵 ∈ 𝐹) → (𝐴 ∪ 𝐵) ⊆ 𝑋)
17		ssun2 4149	. . . . . 6 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
18	17	a1i 11	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐵 ∈ 𝐹) → 𝐵 ⊆ (𝐴 ∪ 𝐵))
19		filss 22461	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐵 ∈ 𝐹 ∧ (𝐴 ∪ 𝐵) ⊆ 𝑋 ∧ 𝐵 ⊆ (𝐴 ∪ 𝐵))) → (𝐴 ∪ 𝐵) ∈ 𝐹)
20	14, 15, 16, 18, 19	syl13anc 1368	. . . 4 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐵 ∈ 𝐹) → (𝐴 ∪ 𝐵) ∈ 𝐹)
21	20	ex 415	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐵 ∈ 𝐹 → (𝐴 ∪ 𝐵) ∈ 𝐹))
22	13, 21	jaod 855	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((𝐴 ∈ 𝐹 ∨ 𝐵 ∈ 𝐹) → (𝐴 ∪ 𝐵) ∈ 𝐹))
23		ufilb 22514	. . . . . . 7 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (¬ 𝐴 ∈ 𝐹 ↔ (𝑋 ∖ 𝐴) ∈ 𝐹))
24	23	3adant3 1128	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (¬ 𝐴 ∈ 𝐹 ↔ (𝑋 ∖ 𝐴) ∈ 𝐹))
25	24	adantr 483	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹) → (¬ 𝐴 ∈ 𝐹 ↔ (𝑋 ∖ 𝐴) ∈ 𝐹))
26	2	3ad2ant1 1129	. . . . . . 7 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹 ∧ (𝑋 ∖ 𝐴) ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
27		difun2 4429	. . . . . . . . . . 11 ⊢ ((𝐵 ∪ 𝐴) ∖ 𝐴) = (𝐵 ∖ 𝐴)
28		uncom 4129	. . . . . . . . . . . 12 ⊢ (𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵)
29	28	difeq1i 4095	. . . . . . . . . . 11 ⊢ ((𝐵 ∪ 𝐴) ∖ 𝐴) = ((𝐴 ∪ 𝐵) ∖ 𝐴)
30	27, 29	eqtr3i 2846	. . . . . . . . . 10 ⊢ (𝐵 ∖ 𝐴) = ((𝐴 ∪ 𝐵) ∖ 𝐴)
31	30	ineq2i 4186	. . . . . . . . 9 ⊢ (𝑋 ∩ (𝐵 ∖ 𝐴)) = (𝑋 ∩ ((𝐴 ∪ 𝐵) ∖ 𝐴))
32		indifcom 4249	. . . . . . . . 9 ⊢ (𝐵 ∩ (𝑋 ∖ 𝐴)) = (𝑋 ∩ (𝐵 ∖ 𝐴))
33		indifcom 4249	. . . . . . . . 9 ⊢ ((𝐴 ∪ 𝐵) ∩ (𝑋 ∖ 𝐴)) = (𝑋 ∩ ((𝐴 ∪ 𝐵) ∖ 𝐴))
34	31, 32, 33	3eqtr4i 2854	. . . . . . . 8 ⊢ (𝐵 ∩ (𝑋 ∖ 𝐴)) = ((𝐴 ∪ 𝐵) ∩ (𝑋 ∖ 𝐴))
35		filin 22462	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹 ∧ (𝑋 ∖ 𝐴) ∈ 𝐹) → ((𝐴 ∪ 𝐵) ∩ (𝑋 ∖ 𝐴)) ∈ 𝐹)
36	2, 35	syl3an1 1159	. . . . . . . 8 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹 ∧ (𝑋 ∖ 𝐴) ∈ 𝐹) → ((𝐴 ∪ 𝐵) ∩ (𝑋 ∖ 𝐴)) ∈ 𝐹)
37	34, 36	eqeltrid 2917	. . . . . . 7 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹 ∧ (𝑋 ∖ 𝐴) ∈ 𝐹) → (𝐵 ∩ (𝑋 ∖ 𝐴)) ∈ 𝐹)
38		simp13 1201	. . . . . . 7 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹 ∧ (𝑋 ∖ 𝐴) ∈ 𝐹) → 𝐵 ⊆ 𝑋)
39		inss1 4205	. . . . . . . 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 22461	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ((𝐵 ∩ (𝑋 ∖ 𝐴)) ∈ 𝐹 ∧ 𝐵 ⊆ 𝑋 ∧ (𝐵 ∩ (𝑋 ∖ 𝐴)) ⊆ 𝐵)) → 𝐵 ∈ 𝐹)
42	26, 37, 38, 40, 41	syl13anc 1368	. . . . . 6 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹 ∧ (𝑋 ∖ 𝐴) ∈ 𝐹) → 𝐵 ∈ 𝐹)
43	42	3expia 1117	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹) → ((𝑋 ∖ 𝐴) ∈ 𝐹 → 𝐵 ∈ 𝐹))
44	25, 43	sylbid 242	. . . 4 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹) → (¬ 𝐴 ∈ 𝐹 → 𝐵 ∈ 𝐹))
45	44	orrd 859	. . 3 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹) → (𝐴 ∈ 𝐹 ∨ 𝐵 ∈ 𝐹))
46	45	ex 415	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((𝐴 ∪ 𝐵) ∈ 𝐹 → (𝐴 ∈ 𝐹 ∨ 𝐵 ∈ 𝐹)))
47	22, 46	impbid 214	1 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((𝐴 ∈ 𝐹 ∨ 𝐵 ∈ 𝐹) ↔ (𝐴 ∪ 𝐵) ∈ 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∧ w3a 1083   ∈ wcel 2114   ∖ cdif 3933   ∪ cun 3934   ∩ cin 3935   ⊆ wss 3936  ‘cfv 6355  Filcfil 22453  UFilcufil 22507

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fv 6363  df-fbas 20542  df-fil 22454  df-ufil 22509

This theorem is referenced by: (None)