Theorem dcun 3560

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 3330	. . . . 5 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ (𝐴 ∪ 𝐵))
2	1	adantl 277	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ (𝐴 ∪ 𝐵))
3	2	orcd 734	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶 ∈ (𝐴 ∪ 𝐵) ∨ ¬ 𝐶 ∈ (𝐴 ∪ 𝐵)))
4		df-dc 836	. . 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 3331	. . . . . 6 ⊢ (𝐶 ∈ 𝐵 → 𝐶 ∈ (𝐴 ∪ 𝐵))
7	6	adantl 277	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ 𝐴) ∧ 𝐶 ∈ 𝐵) → 𝐶 ∈ (𝐴 ∪ 𝐵))
8	7	orcd 734	. . . 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 753	. . . . . . 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 3304	. . . . . 6 ⊢ (𝐶 ∈ (𝐴 ∪ 𝐵) ↔ (𝐶 ∈ 𝐴 ∨ 𝐶 ∈ 𝐵))
15	13, 14	sylnibr 678	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ 𝐴) ∧ ¬ 𝐶 ∈ 𝐵) → ¬ 𝐶 ∈ (𝐴 ∪ 𝐵))
16	15	olcd 735	. . . 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 837	. . . . 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 799	. 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝐴) → DECID 𝐶 ∈ (𝐴 ∪ 𝐵))
23		dcun.a	. . 3 ⊢ (𝜑 → DECID 𝐶 ∈ 𝐴)
24		exmiddc 837	. . 3 ⊢ (DECID 𝐶 ∈ 𝐴 → (𝐶 ∈ 𝐴 ∨ ¬ 𝐶 ∈ 𝐴))
25	23, 24	syl 14	. 2 ⊢ (𝜑 → (𝐶 ∈ 𝐴 ∨ ¬ 𝐶 ∈ 𝐴))
26	5, 22, 25	mpjaodan 799	1 ⊢ (𝜑 → DECID 𝐶 ∈ (𝐴 ∪ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 709  DECID wdc 835   ∈ wcel 2167   ∪ cun 3155

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170

This theorem is referenced by:  tpfidceq  6991  sumsplitdc  11597