Theorem dcun 3531

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 3300	. . . . 5 ⊢ (𝑘 ∈ 𝐴 → 𝑘 ∈ (𝐴 ∪ 𝐵))
2	1	adantl 277	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ (𝐴 ∪ 𝐵))
3	2	orcd 733	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑘 ∈ (𝐴 ∪ 𝐵) ∨ ¬ 𝑘 ∈ (𝐴 ∪ 𝐵)))
4		df-dc 835	. . 3 ⊢ (DECID 𝑘 ∈ (𝐴 ∪ 𝐵) ↔ (𝑘 ∈ (𝐴 ∪ 𝐵) ∨ ¬ 𝑘 ∈ (𝐴 ∪ 𝐵)))
5	3, 4	sylibr 134	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → DECID 𝑘 ∈ (𝐴 ∪ 𝐵))
6		elun2 3301	. . . . . 6 ⊢ (𝑘 ∈ 𝐵 → 𝑘 ∈ (𝐴 ∪ 𝐵))
7	6	adantl 277	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝑘 ∈ (𝐴 ∪ 𝐵))
8	7	orcd 733	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → (𝑘 ∈ (𝐴 ∪ 𝐵) ∨ ¬ 𝑘 ∈ (𝐴 ∪ 𝐵)))
9	8, 4	sylibr 134	. . 3 ⊢ (((𝜑 ∧ ¬ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → DECID 𝑘 ∈ (𝐴 ∪ 𝐵))
10		simplr 528	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑘 ∈ 𝐴) ∧ ¬ 𝑘 ∈ 𝐵) → ¬ 𝑘 ∈ 𝐴)
11		simpr 110	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑘 ∈ 𝐴) ∧ ¬ 𝑘 ∈ 𝐵) → ¬ 𝑘 ∈ 𝐵)
12		ioran 752	. . . . . . 7 ⊢ (¬ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵) ↔ (¬ 𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ 𝐵))
13	10, 11, 12	sylanbrc 417	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑘 ∈ 𝐴) ∧ ¬ 𝑘 ∈ 𝐵) → ¬ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵))
14		elun 3274	. . . . . 6 ⊢ (𝑘 ∈ (𝐴 ∪ 𝐵) ↔ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵))
15	13, 14	sylnibr 677	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑘 ∈ 𝐴) ∧ ¬ 𝑘 ∈ 𝐵) → ¬ 𝑘 ∈ (𝐴 ∪ 𝐵))
16	15	olcd 734	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑘 ∈ 𝐴) ∧ ¬ 𝑘 ∈ 𝐵) → (𝑘 ∈ (𝐴 ∪ 𝐵) ∨ ¬ 𝑘 ∈ (𝐴 ∪ 𝐵)))
17	16, 4	sylibr 134	. . 3 ⊢ (((𝜑 ∧ ¬ 𝑘 ∈ 𝐴) ∧ ¬ 𝑘 ∈ 𝐵) → DECID 𝑘 ∈ (𝐴 ∪ 𝐵))
18		dcun.b	. . . . 5 ⊢ (𝜑 → DECID 𝑘 ∈ 𝐵)
19		exmiddc 836	. . . . 5 ⊢ (DECID 𝑘 ∈ 𝐵 → (𝑘 ∈ 𝐵 ∨ ¬ 𝑘 ∈ 𝐵))
20	18, 19	syl 14	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐵 ∨ ¬ 𝑘 ∈ 𝐵))
21	20	adantr 276	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑘 ∈ 𝐴) → (𝑘 ∈ 𝐵 ∨ ¬ 𝑘 ∈ 𝐵))
22	9, 17, 21	mpjaodan 798	. 2 ⊢ ((𝜑 ∧ ¬ 𝑘 ∈ 𝐴) → DECID 𝑘 ∈ (𝐴 ∪ 𝐵))
23		dcun.a	. . 3 ⊢ (𝜑 → DECID 𝑘 ∈ 𝐴)
24		exmiddc 836	. . 3 ⊢ (DECID 𝑘 ∈ 𝐴 → (𝑘 ∈ 𝐴 ∨ ¬ 𝑘 ∈ 𝐴))
25	23, 24	syl 14	. 2 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ∨ ¬ 𝑘 ∈ 𝐴))
26	5, 22, 25	mpjaodan 798	1 ⊢ (𝜑 → DECID 𝑘 ∈ (𝐴 ∪ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 708  DECID wdc 834   ∈ wcel 2146   ∪ cun 3125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-dc 835  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-un 3131  df-in 3133  df-ss 3140

This theorem is referenced by:  sumsplitdc  11406