Theorem ufprim 24255

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 24250	. . . . . . 7 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
2	1	3ad2ant1 1151	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → 𝐹 ∈ (Fil‘𝑋))
3	2	adantr 486	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
4		simpr 490	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ∈ 𝐹)
5		unss 4191	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ↔ (𝐴 ∪ 𝐵) ⊆ 𝑋)
6	5	biimpi 219	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐴 ∪ 𝐵) ⊆ 𝑋)
7	6	3adant1 1148	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐴 ∪ 𝐵) ⊆ 𝑋)
8	7	adantr 486	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐴 ∪ 𝐵) ⊆ 𝑋)
9		ssun1 4179	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
10	9	a1i 11	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ (𝐴 ∪ 𝐵))
11		filss 24199	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∈ 𝐹 ∧ (𝐴 ∪ 𝐵) ⊆ 𝑋 ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵))) → (𝐴 ∪ 𝐵) ∈ 𝐹)
12	3, 4, 8, 10, 11	syl13anc 1400	. . . 4 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐴 ∪ 𝐵) ∈ 𝐹)
13	12	ex 418	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐴 ∈ 𝐹 → (𝐴 ∪ 𝐵) ∈ 𝐹))
14	2	adantr 486	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐵 ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
15		simpr 490	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐵 ∈ 𝐹) → 𝐵 ∈ 𝐹)
16	7	adantr 486	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐵 ∈ 𝐹) → (𝐴 ∪ 𝐵) ⊆ 𝑋)
17		ssun2 4180	. . . . . 6 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
18	17	a1i 11	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐵 ∈ 𝐹) → 𝐵 ⊆ (𝐴 ∪ 𝐵))
19		filss 24199	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐵 ∈ 𝐹 ∧ (𝐴 ∪ 𝐵) ⊆ 𝑋 ∧ 𝐵 ⊆ (𝐴 ∪ 𝐵))) → (𝐴 ∪ 𝐵) ∈ 𝐹)
20	14, 15, 16, 18, 19	syl13anc 1400	. . . 4 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐵 ∈ 𝐹) → (𝐴 ∪ 𝐵) ∈ 𝐹)
21	20	ex 418	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐵 ∈ 𝐹 → (𝐴 ∪ 𝐵) ∈ 𝐹))
22	13, 21	jaod 873	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((𝐴 ∈ 𝐹 ∨ 𝐵 ∈ 𝐹) → (𝐴 ∪ 𝐵) ∈ 𝐹))
23		ufilb 24252	. . . . . . 7 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (¬ 𝐴 ∈ 𝐹 ↔ (𝑋 ∖ 𝐴) ∈ 𝐹))
24	23	3adant3 1150	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (¬ 𝐴 ∈ 𝐹 ↔ (𝑋 ∖ 𝐴) ∈ 𝐹))
25	24	adantr 486	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹) → (¬ 𝐴 ∈ 𝐹 ↔ (𝑋 ∖ 𝐴) ∈ 𝐹))
26	2	3ad2ant1 1151	. . . . . . 7 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹 ∧ (𝑋 ∖ 𝐴) ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
27		difun2 4440	. . . . . . . . . . 11 ⊢ ((𝐵 ∪ 𝐴) ∖ 𝐴) = (𝐵 ∖ 𝐴)
28		uncom 4160	. . . . . . . . . . . 12 ⊢ (𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵)
29	28	difeq1i 4062	. . . . . . . . . . 11 ⊢ ((𝐵 ∪ 𝐴) ∖ 𝐴) = ((𝐴 ∪ 𝐵) ∖ 𝐴)
30	27, 29	eqtr3i 2788	. . . . . . . . . 10 ⊢ (𝐵 ∖ 𝐴) = ((𝐴 ∪ 𝐵) ∖ 𝐴)
31	30	ineq2i 4117	. . . . . . . . 9 ⊢ (𝑋 ∩ (𝐵 ∖ 𝐴)) = (𝑋 ∩ ((𝐴 ∪ 𝐵) ∖ 𝐴))
32		indifcom 4231	. . . . . . . . 9 ⊢ (𝐵 ∩ (𝑋 ∖ 𝐴)) = (𝑋 ∩ (𝐵 ∖ 𝐴))
33		indifcom 4231	. . . . . . . . 9 ⊢ ((𝐴 ∪ 𝐵) ∩ (𝑋 ∖ 𝐴)) = (𝑋 ∩ ((𝐴 ∪ 𝐵) ∖ 𝐴))
34	31, 32, 33	3eqtr4i 2796	. . . . . . . 8 ⊢ (𝐵 ∩ (𝑋 ∖ 𝐴)) = ((𝐴 ∪ 𝐵) ∩ (𝑋 ∖ 𝐴))
35		filin 24200	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹 ∧ (𝑋 ∖ 𝐴) ∈ 𝐹) → ((𝐴 ∪ 𝐵) ∩ (𝑋 ∖ 𝐴)) ∈ 𝐹)
36	2, 35	syl3an1 1181	. . . . . . . 8 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹 ∧ (𝑋 ∖ 𝐴) ∈ 𝐹) → ((𝐴 ∪ 𝐵) ∩ (𝑋 ∖ 𝐴)) ∈ 𝐹)
37	34, 36	eqeltrid 2868	. . . . . . 7 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹 ∧ (𝑋 ∖ 𝐴) ∈ 𝐹) → (𝐵 ∩ (𝑋 ∖ 𝐴)) ∈ 𝐹)
38		simp13 1224	. . . . . . 7 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹 ∧ (𝑋 ∖ 𝐴) ∈ 𝐹) → 𝐵 ⊆ 𝑋)
39		inss1 4136	. . . . . . . 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 24199	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ((𝐵 ∩ (𝑋 ∖ 𝐴)) ∈ 𝐹 ∧ 𝐵 ⊆ 𝑋 ∧ (𝐵 ∩ (𝑋 ∖ 𝐴)) ⊆ 𝐵)) → 𝐵 ∈ 𝐹)
42	26, 37, 38, 40, 41	syl13anc 1400	. . . . . 6 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹 ∧ (𝑋 ∖ 𝐴) ∈ 𝐹) → 𝐵 ∈ 𝐹)
43	42	3expia 1139	. . . . 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 877	. . 3 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹) → (𝐴 ∈ 𝐹 ∨ 𝐵 ∈ 𝐹))
46	45	ex 418	. 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))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   ∈ wcel 2147   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092   ∪ cun 4152  ‘cfv 6555  Filcfil 24191  UFilcufil 24245

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fv 6563  df-fbas 21702  df-fil 24192  df-ufil 24247

This theorem is used by: (None)