Theorem dcun 3581

Description: The union of two decidable classes is decidable. (Contributed by Jim Kingdon, 5-Oct-2022.) (Revised by Jim Kingdon, 13-Oct-2025.)

Hypotheses

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

Assertion

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

Proof of Theorem dcun

Step	Hyp	Ref	Expression
1		elun1 3351	. . . . 5 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ (𝐴 ∪ 𝐵))
2	1	adantl 277	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ (𝐴 ∪ 𝐵))
3	2	orcd 737	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶 ∈ (𝐴 ∪ 𝐵) ∨ ¬ 𝐶 ∈ (𝐴 ∪ 𝐵)))
4		df-dc 839	. . 3 ⊢ (DECID 𝐶 ∈ (𝐴 ∪ 𝐵) ↔ (𝐶 ∈ (𝐴 ∪ 𝐵) ∨ ¬ 𝐶 ∈ (𝐴 ∪ 𝐵)))
5	3, 4	sylibr 134	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → DECID 𝐶 ∈ (𝐴 ∪ 𝐵))
6		elun2 3352	. . . . . 6 ⊢ (𝐶 ∈ 𝐵 → 𝐶 ∈ (𝐴 ∪ 𝐵))
7	6	adantl 277	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ 𝐴) ∧ 𝐶 ∈ 𝐵) → 𝐶 ∈ (𝐴 ∪ 𝐵))
8	7	orcd 737	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ 𝐴) ∧ 𝐶 ∈ 𝐵) → (𝐶 ∈ (𝐴 ∪ 𝐵) ∨ ¬ 𝐶 ∈ (𝐴 ∪ 𝐵)))
9	8, 4	sylibr 134	. . 3 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ 𝐴) ∧ 𝐶 ∈ 𝐵) → DECID 𝐶 ∈ (𝐴 ∪ 𝐵))
10		simplr 528	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ 𝐴) ∧ ¬ 𝐶 ∈ 𝐵) → ¬ 𝐶 ∈ 𝐴)
11		simpr 110	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ 𝐴) ∧ ¬ 𝐶 ∈ 𝐵) → ¬ 𝐶 ∈ 𝐵)
12		ioran 756	. . . . . . 7 ⊢ (¬ (𝐶 ∈ 𝐴 ∨ 𝐶 ∈ 𝐵) ↔ (¬ 𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐵))
13	10, 11, 12	sylanbrc 417	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ 𝐴) ∧ ¬ 𝐶 ∈ 𝐵) → ¬ (𝐶 ∈ 𝐴 ∨ 𝐶 ∈ 𝐵))
14		elun 3325	. . . . . 6 ⊢ (𝐶 ∈ (𝐴 ∪ 𝐵) ↔ (𝐶 ∈ 𝐴 ∨ 𝐶 ∈ 𝐵))
15	13, 14	sylnibr 681	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ 𝐴) ∧ ¬ 𝐶 ∈ 𝐵) → ¬ 𝐶 ∈ (𝐴 ∪ 𝐵))
16	15	olcd 738	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ 𝐴) ∧ ¬ 𝐶 ∈ 𝐵) → (𝐶 ∈ (𝐴 ∪ 𝐵) ∨ ¬ 𝐶 ∈ (𝐴 ∪ 𝐵)))
17	16, 4	sylibr 134	. . 3 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ 𝐴) ∧ ¬ 𝐶 ∈ 𝐵) → DECID 𝐶 ∈ (𝐴 ∪ 𝐵))
18		dcun.b	. . . . 5 ⊢ (𝜑 → DECID 𝐶 ∈ 𝐵)
19		exmiddc 840	. . . . 5 ⊢ (DECID 𝐶 ∈ 𝐵 → (𝐶 ∈ 𝐵 ∨ ¬ 𝐶 ∈ 𝐵))
20	18, 19	syl 14	. . . 4 ⊢ (𝜑 → (𝐶 ∈ 𝐵 ∨ ¬ 𝐶 ∈ 𝐵))
21	20	adantr 276	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝐴) → (𝐶 ∈ 𝐵 ∨ ¬ 𝐶 ∈ 𝐵))
22	9, 17, 21	mpjaodan 802	. 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝐴) → DECID 𝐶 ∈ (𝐴 ∪ 𝐵))
23		dcun.a	. . 3 ⊢ (𝜑 → DECID 𝐶 ∈ 𝐴)
24		exmiddc 840	. . 3 ⊢ (DECID 𝐶 ∈ 𝐴 → (𝐶 ∈ 𝐴 ∨ ¬ 𝐶 ∈ 𝐴))
25	23, 24	syl 14	. 2 ⊢ (𝜑 → (𝐶 ∈ 𝐴 ∨ ¬ 𝐶 ∈ 𝐴))
26	5, 22, 25	mpjaodan 802	1 ⊢ (𝜑 → DECID 𝐶 ∈ (𝐴 ∪ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 712  DECID wdc 838   ∈ wcel 2180   ∪ cun 3175

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-dc 839  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-v 2781  df-un 3181  df-in 3183  df-ss 3190

This theorem is referenced by:  tpfidceq  7060  sumsplitdc  11909