Theorem ufprim 22205

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 22200	. . . . . . 7 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
2	1	3ad2ant1 1126	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → 𝐹 ∈ (Fil‘𝑋))
3	2	adantr 481	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
4		simpr 485	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ∈ 𝐹)
5		unss 4087	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ↔ (𝐴 ∪ 𝐵) ⊆ 𝑋)
6	5	biimpi 217	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐴 ∪ 𝐵) ⊆ 𝑋)
7	6	3adant1 1123	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐴 ∪ 𝐵) ⊆ 𝑋)
8	7	adantr 481	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐴 ∪ 𝐵) ⊆ 𝑋)
9		ssun1 4075	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
10	9	a1i 11	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ (𝐴 ∪ 𝐵))
11		filss 22149	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∈ 𝐹 ∧ (𝐴 ∪ 𝐵) ⊆ 𝑋 ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵))) → (𝐴 ∪ 𝐵) ∈ 𝐹)
12	3, 4, 8, 10, 11	syl13anc 1365	. . . 4 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐴 ∪ 𝐵) ∈ 𝐹)
13	12	ex 413	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐴 ∈ 𝐹 → (𝐴 ∪ 𝐵) ∈ 𝐹))
14	2	adantr 481	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐵 ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
15		simpr 485	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐵 ∈ 𝐹) → 𝐵 ∈ 𝐹)
16	7	adantr 481	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐵 ∈ 𝐹) → (𝐴 ∪ 𝐵) ⊆ 𝑋)
17		ssun2 4076	. . . . . 6 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
18	17	a1i 11	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐵 ∈ 𝐹) → 𝐵 ⊆ (𝐴 ∪ 𝐵))
19		filss 22149	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐵 ∈ 𝐹 ∧ (𝐴 ∪ 𝐵) ⊆ 𝑋 ∧ 𝐵 ⊆ (𝐴 ∪ 𝐵))) → (𝐴 ∪ 𝐵) ∈ 𝐹)
20	14, 15, 16, 18, 19	syl13anc 1365	. . . 4 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ 𝐵 ∈ 𝐹) → (𝐴 ∪ 𝐵) ∈ 𝐹)
21	20	ex 413	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐵 ∈ 𝐹 → (𝐴 ∪ 𝐵) ∈ 𝐹))
22	13, 21	jaod 854	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((𝐴 ∈ 𝐹 ∨ 𝐵 ∈ 𝐹) → (𝐴 ∪ 𝐵) ∈ 𝐹))
23		ufilb 22202	. . . . . . 7 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (¬ 𝐴 ∈ 𝐹 ↔ (𝑋 ∖ 𝐴) ∈ 𝐹))
24	23	3adant3 1125	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (¬ 𝐴 ∈ 𝐹 ↔ (𝑋 ∖ 𝐴) ∈ 𝐹))
25	24	adantr 481	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹) → (¬ 𝐴 ∈ 𝐹 ↔ (𝑋 ∖ 𝐴) ∈ 𝐹))
26	2	3ad2ant1 1126	. . . . . . 7 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹 ∧ (𝑋 ∖ 𝐴) ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
27		difun2 4349	. . . . . . . . . . 11 ⊢ ((𝐵 ∪ 𝐴) ∖ 𝐴) = (𝐵 ∖ 𝐴)
28		uncom 4056	. . . . . . . . . . . 12 ⊢ (𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵)
29	28	difeq1i 4022	. . . . . . . . . . 11 ⊢ ((𝐵 ∪ 𝐴) ∖ 𝐴) = ((𝐴 ∪ 𝐵) ∖ 𝐴)
30	27, 29	eqtr3i 2823	. . . . . . . . . 10 ⊢ (𝐵 ∖ 𝐴) = ((𝐴 ∪ 𝐵) ∖ 𝐴)
31	30	ineq2i 4112	. . . . . . . . 9 ⊢ (𝑋 ∩ (𝐵 ∖ 𝐴)) = (𝑋 ∩ ((𝐴 ∪ 𝐵) ∖ 𝐴))
32		indifcom 4175	. . . . . . . . 9 ⊢ (𝐵 ∩ (𝑋 ∖ 𝐴)) = (𝑋 ∩ (𝐵 ∖ 𝐴))
33		indifcom 4175	. . . . . . . . 9 ⊢ ((𝐴 ∪ 𝐵) ∩ (𝑋 ∖ 𝐴)) = (𝑋 ∩ ((𝐴 ∪ 𝐵) ∖ 𝐴))
34	31, 32, 33	3eqtr4i 2831	. . . . . . . 8 ⊢ (𝐵 ∩ (𝑋 ∖ 𝐴)) = ((𝐴 ∪ 𝐵) ∩ (𝑋 ∖ 𝐴))
35		filin 22150	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹 ∧ (𝑋 ∖ 𝐴) ∈ 𝐹) → ((𝐴 ∪ 𝐵) ∩ (𝑋 ∖ 𝐴)) ∈ 𝐹)
36	2, 35	syl3an1 1156	. . . . . . . 8 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹 ∧ (𝑋 ∖ 𝐴) ∈ 𝐹) → ((𝐴 ∪ 𝐵) ∩ (𝑋 ∖ 𝐴)) ∈ 𝐹)
37	34, 36	syl5eqel 2889	. . . . . . 7 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹 ∧ (𝑋 ∖ 𝐴) ∈ 𝐹) → (𝐵 ∩ (𝑋 ∖ 𝐴)) ∈ 𝐹)
38		simp13 1198	. . . . . . 7 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹 ∧ (𝑋 ∖ 𝐴) ∈ 𝐹) → 𝐵 ⊆ 𝑋)
39		inss1 4131	. . . . . . . 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 22149	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ((𝐵 ∩ (𝑋 ∖ 𝐴)) ∈ 𝐹 ∧ 𝐵 ⊆ 𝑋 ∧ (𝐵 ∩ (𝑋 ∖ 𝐴)) ⊆ 𝐵)) → 𝐵 ∈ 𝐹)
42	26, 37, 38, 40, 41	syl13anc 1365	. . . . . 6 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹 ∧ (𝑋 ∖ 𝐴) ∈ 𝐹) → 𝐵 ∈ 𝐹)
43	42	3expia 1114	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹) → ((𝑋 ∖ 𝐴) ∈ 𝐹 → 𝐵 ∈ 𝐹))
44	25, 43	sylbid 241	. . . 4 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹) → (¬ 𝐴 ∈ 𝐹 → 𝐵 ∈ 𝐹))
45	44	orrd 858	. . 3 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∪ 𝐵) ∈ 𝐹) → (𝐴 ∈ 𝐹 ∨ 𝐵 ∈ 𝐹))
46	45	ex 413	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((𝐴 ∪ 𝐵) ∈ 𝐹 → (𝐴 ∈ 𝐹 ∨ 𝐵 ∈ 𝐹)))
47	22, 46	impbid 213	1 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((𝐴 ∈ 𝐹 ∨ 𝐵 ∈ 𝐹) ↔ (𝐴 ∪ 𝐵) ∈ 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 842   ∧ w3a 1080   ∈ wcel 2083   ∖ cdif 3862   ∪ cun 3863   ∩ cin 3864   ⊆ wss 3865  ‘cfv 6232  Filcfil 22141  UFilcufil 22195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fv 6240  df-fbas 20228  df-fil 22142  df-ufil 22197

This theorem is referenced by: (None)