Theorem dcun 3571

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 3341	. . . . 5 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ (𝐴 ∪ 𝐵))
2	1	adantl 277	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ (𝐴 ∪ 𝐵))
3	2	orcd 735	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶 ∈ (𝐴 ∪ 𝐵) ∨ ¬ 𝐶 ∈ (𝐴 ∪ 𝐵)))
4		df-dc 837	. . 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 3342	. . . . . 6 ⊢ (𝐶 ∈ 𝐵 → 𝐶 ∈ (𝐴 ∪ 𝐵))
7	6	adantl 277	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ 𝐴) ∧ 𝐶 ∈ 𝐵) → 𝐶 ∈ (𝐴 ∪ 𝐵))
8	7	orcd 735	. . . 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 754	. . . . . . 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 3315	. . . . . 6 ⊢ (𝐶 ∈ (𝐴 ∪ 𝐵) ↔ (𝐶 ∈ 𝐴 ∨ 𝐶 ∈ 𝐵))
15	13, 14	sylnibr 679	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ 𝐴) ∧ ¬ 𝐶 ∈ 𝐵) → ¬ 𝐶 ∈ (𝐴 ∪ 𝐵))
16	15	olcd 736	. . . 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 838	. . . . 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 800	. 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝐴) → DECID 𝐶 ∈ (𝐴 ∪ 𝐵))
23		dcun.a	. . 3 ⊢ (𝜑 → DECID 𝐶 ∈ 𝐴)
24		exmiddc 838	. . 3 ⊢ (DECID 𝐶 ∈ 𝐴 → (𝐶 ∈ 𝐴 ∨ ¬ 𝐶 ∈ 𝐴))
25	23, 24	syl 14	. 2 ⊢ (𝜑 → (𝐶 ∈ 𝐴 ∨ ¬ 𝐶 ∈ 𝐴))
26	5, 22, 25	mpjaodan 800	1 ⊢ (𝜑 → DECID 𝐶 ∈ (𝐴 ∪ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 710  DECID wdc 836   ∈ wcel 2177   ∪ cun 3165

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-dc 837  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3171  df-in 3173  df-ss 3180

This theorem is referenced by:  tpfidceq  7034  sumsplitdc  11787