Theorem dcun 3556

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 3326	. . . . 5 ⊢ (𝑘 ∈ 𝐴 → 𝑘 ∈ (𝐴 ∪ 𝐵))
2	1	adantl 277	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ (𝐴 ∪ 𝐵))
3	2	orcd 734	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑘 ∈ (𝐴 ∪ 𝐵) ∨ ¬ 𝑘 ∈ (𝐴 ∪ 𝐵)))
4		df-dc 836	. . 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 3327	. . . . . 6 ⊢ (𝑘 ∈ 𝐵 → 𝑘 ∈ (𝐴 ∪ 𝐵))
7	6	adantl 277	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝑘 ∈ (𝐴 ∪ 𝐵))
8	7	orcd 734	. . . 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 753	. . . . . . 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 3300	. . . . . 6 ⊢ (𝑘 ∈ (𝐴 ∪ 𝐵) ↔ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵))
15	13, 14	sylnibr 678	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑘 ∈ 𝐴) ∧ ¬ 𝑘 ∈ 𝐵) → ¬ 𝑘 ∈ (𝐴 ∪ 𝐵))
16	15	olcd 735	. . . 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 837	. . . . 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 799	. 2 ⊢ ((𝜑 ∧ ¬ 𝑘 ∈ 𝐴) → DECID 𝑘 ∈ (𝐴 ∪ 𝐵))
23		dcun.a	. . 3 ⊢ (𝜑 → DECID 𝑘 ∈ 𝐴)
24		exmiddc 837	. . 3 ⊢ (DECID 𝑘 ∈ 𝐴 → (𝑘 ∈ 𝐴 ∨ ¬ 𝑘 ∈ 𝐴))
25	23, 24	syl 14	. 2 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ∨ ¬ 𝑘 ∈ 𝐴))
26	5, 22, 25	mpjaodan 799	1 ⊢ (𝜑 → DECID 𝑘 ∈ (𝐴 ∪ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 709  DECID wdc 835   ∈ wcel 2164   ∪ cun 3151

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166

This theorem is referenced by:  sumsplitdc  11575