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Theorem dcun 3473
Description: The union of two decidable classes is decidable. (Contributed by Jim Kingdon, 5-Oct-2022.)
Hypotheses
Ref Expression
dcun.a (𝜑DECID 𝑘𝐴)
dcun.b (𝜑DECID 𝑘𝐵)
Assertion
Ref Expression
dcun (𝜑DECID 𝑘 ∈ (𝐴𝐵))

Proof of Theorem dcun
StepHypRef Expression
1 elun1 3243 . . . . 5 (𝑘𝐴𝑘 ∈ (𝐴𝐵))
21adantl 275 . . . 4 ((𝜑𝑘𝐴) → 𝑘 ∈ (𝐴𝐵))
32orcd 722 . . 3 ((𝜑𝑘𝐴) → (𝑘 ∈ (𝐴𝐵) ∨ ¬ 𝑘 ∈ (𝐴𝐵)))
4 df-dc 820 . . 3 (DECID 𝑘 ∈ (𝐴𝐵) ↔ (𝑘 ∈ (𝐴𝐵) ∨ ¬ 𝑘 ∈ (𝐴𝐵)))
53, 4sylibr 133 . 2 ((𝜑𝑘𝐴) → DECID 𝑘 ∈ (𝐴𝐵))
6 elun2 3244 . . . . . 6 (𝑘𝐵𝑘 ∈ (𝐴𝐵))
76adantl 275 . . . . 5 (((𝜑 ∧ ¬ 𝑘𝐴) ∧ 𝑘𝐵) → 𝑘 ∈ (𝐴𝐵))
87orcd 722 . . . 4 (((𝜑 ∧ ¬ 𝑘𝐴) ∧ 𝑘𝐵) → (𝑘 ∈ (𝐴𝐵) ∨ ¬ 𝑘 ∈ (𝐴𝐵)))
98, 4sylibr 133 . . 3 (((𝜑 ∧ ¬ 𝑘𝐴) ∧ 𝑘𝐵) → DECID 𝑘 ∈ (𝐴𝐵))
10 simplr 519 . . . . . . 7 (((𝜑 ∧ ¬ 𝑘𝐴) ∧ ¬ 𝑘𝐵) → ¬ 𝑘𝐴)
11 simpr 109 . . . . . . 7 (((𝜑 ∧ ¬ 𝑘𝐴) ∧ ¬ 𝑘𝐵) → ¬ 𝑘𝐵)
12 ioran 741 . . . . . . 7 (¬ (𝑘𝐴𝑘𝐵) ↔ (¬ 𝑘𝐴 ∧ ¬ 𝑘𝐵))
1310, 11, 12sylanbrc 413 . . . . . 6 (((𝜑 ∧ ¬ 𝑘𝐴) ∧ ¬ 𝑘𝐵) → ¬ (𝑘𝐴𝑘𝐵))
14 elun 3217 . . . . . 6 (𝑘 ∈ (𝐴𝐵) ↔ (𝑘𝐴𝑘𝐵))
1513, 14sylnibr 666 . . . . 5 (((𝜑 ∧ ¬ 𝑘𝐴) ∧ ¬ 𝑘𝐵) → ¬ 𝑘 ∈ (𝐴𝐵))
1615olcd 723 . . . 4 (((𝜑 ∧ ¬ 𝑘𝐴) ∧ ¬ 𝑘𝐵) → (𝑘 ∈ (𝐴𝐵) ∨ ¬ 𝑘 ∈ (𝐴𝐵)))
1716, 4sylibr 133 . . 3 (((𝜑 ∧ ¬ 𝑘𝐴) ∧ ¬ 𝑘𝐵) → DECID 𝑘 ∈ (𝐴𝐵))
18 dcun.b . . . . 5 (𝜑DECID 𝑘𝐵)
19 exmiddc 821 . . . . 5 (DECID 𝑘𝐵 → (𝑘𝐵 ∨ ¬ 𝑘𝐵))
2018, 19syl 14 . . . 4 (𝜑 → (𝑘𝐵 ∨ ¬ 𝑘𝐵))
2120adantr 274 . . 3 ((𝜑 ∧ ¬ 𝑘𝐴) → (𝑘𝐵 ∨ ¬ 𝑘𝐵))
229, 17, 21mpjaodan 787 . 2 ((𝜑 ∧ ¬ 𝑘𝐴) → DECID 𝑘 ∈ (𝐴𝐵))
23 dcun.a . . 3 (𝜑DECID 𝑘𝐴)
24 exmiddc 821 . . 3 (DECID 𝑘𝐴 → (𝑘𝐴 ∨ ¬ 𝑘𝐴))
2523, 24syl 14 . 2 (𝜑 → (𝑘𝐴 ∨ ¬ 𝑘𝐴))
265, 22, 25mpjaodan 787 1 (𝜑DECID 𝑘 ∈ (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 697  DECID wdc 819  wcel 1480  cun 3069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084
This theorem is referenced by:  sumsplitdc  11215
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