Theorem dcun 3412

Description: The union of two decidable classes is decidable. (Contributed by Jim Kingdon, 5-Oct-2022.)

Hypotheses

Ref	Expression
dcun.a	⊢ (𝜑 → DECID 𝑘 ∈ 𝐴)
dcun.b	⊢ (𝜑 → DECID 𝑘 ∈ 𝐵)

Assertion

Ref	Expression
dcun	⊢ (𝜑 → DECID 𝑘 ∈ (𝐴 ∪ 𝐵))

Proof of Theorem dcun

Step	Hyp	Ref	Expression
1		elun1 3182	. . . . 5 ⊢ (𝑘 ∈ 𝐴 → 𝑘 ∈ (𝐴 ∪ 𝐵))
2	1	adantl 272	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ (𝐴 ∪ 𝐵))
3	2	orcd 690	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑘 ∈ (𝐴 ∪ 𝐵) ∨ ¬ 𝑘 ∈ (𝐴 ∪ 𝐵)))
4		df-dc 784	. . 3 ⊢ (DECID 𝑘 ∈ (𝐴 ∪ 𝐵) ↔ (𝑘 ∈ (𝐴 ∪ 𝐵) ∨ ¬ 𝑘 ∈ (𝐴 ∪ 𝐵)))
5	3, 4	sylibr 133	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → DECID 𝑘 ∈ (𝐴 ∪ 𝐵))
6		elun2 3183	. . . . . 6 ⊢ (𝑘 ∈ 𝐵 → 𝑘 ∈ (𝐴 ∪ 𝐵))
7	6	adantl 272	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝑘 ∈ (𝐴 ∪ 𝐵))
8	7	orcd 690	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → (𝑘 ∈ (𝐴 ∪ 𝐵) ∨ ¬ 𝑘 ∈ (𝐴 ∪ 𝐵)))
9	8, 4	sylibr 133	. . 3 ⊢ (((𝜑 ∧ ¬ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → DECID 𝑘 ∈ (𝐴 ∪ 𝐵))
10		simplr 498	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑘 ∈ 𝐴) ∧ ¬ 𝑘 ∈ 𝐵) → ¬ 𝑘 ∈ 𝐴)
11		simpr 109	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑘 ∈ 𝐴) ∧ ¬ 𝑘 ∈ 𝐵) → ¬ 𝑘 ∈ 𝐵)
12		ioran 707	. . . . . . 7 ⊢ (¬ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵) ↔ (¬ 𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ 𝐵))
13	10, 11, 12	sylanbrc 409	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑘 ∈ 𝐴) ∧ ¬ 𝑘 ∈ 𝐵) → ¬ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵))
14		elun 3156	. . . . . 6 ⊢ (𝑘 ∈ (𝐴 ∪ 𝐵) ↔ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵))
15	13, 14	sylnibr 640	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑘 ∈ 𝐴) ∧ ¬ 𝑘 ∈ 𝐵) → ¬ 𝑘 ∈ (𝐴 ∪ 𝐵))
16	15	olcd 691	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑘 ∈ 𝐴) ∧ ¬ 𝑘 ∈ 𝐵) → (𝑘 ∈ (𝐴 ∪ 𝐵) ∨ ¬ 𝑘 ∈ (𝐴 ∪ 𝐵)))
17	16, 4	sylibr 133	. . 3 ⊢ (((𝜑 ∧ ¬ 𝑘 ∈ 𝐴) ∧ ¬ 𝑘 ∈ 𝐵) → DECID 𝑘 ∈ (𝐴 ∪ 𝐵))
18		dcun.b	. . . . 5 ⊢ (𝜑 → DECID 𝑘 ∈ 𝐵)
19		exmiddc 785	. . . . 5 ⊢ (DECID 𝑘 ∈ 𝐵 → (𝑘 ∈ 𝐵 ∨ ¬ 𝑘 ∈ 𝐵))
20	18, 19	syl 14	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐵 ∨ ¬ 𝑘 ∈ 𝐵))
21	20	adantr 271	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑘 ∈ 𝐴) → (𝑘 ∈ 𝐵 ∨ ¬ 𝑘 ∈ 𝐵))
22	9, 17, 21	mpjaodan 750	. 2 ⊢ ((𝜑 ∧ ¬ 𝑘 ∈ 𝐴) → DECID 𝑘 ∈ (𝐴 ∪ 𝐵))
23		dcun.a	. . 3 ⊢ (𝜑 → DECID 𝑘 ∈ 𝐴)
24		exmiddc 785	. . 3 ⊢ (DECID 𝑘 ∈ 𝐴 → (𝑘 ∈ 𝐴 ∨ ¬ 𝑘 ∈ 𝐴))
25	23, 24	syl 14	. 2 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ∨ ¬ 𝑘 ∈ 𝐴))
26	5, 22, 25	mpjaodan 750	1 ⊢ (𝜑 → DECID 𝑘 ∈ (𝐴 ∪ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 667  DECID wdc 783   ∈ wcel 1445   ∪ cun 3011

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-dc 784  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-un 3017  df-in 3019  df-ss 3026

This theorem is referenced by:  sumsplitdc  10975