Theorem decidin 13678

Description: If A is a decidable subclass of B (meaning: it is a subclass of B and it is decidable in B), and B is decidable in C, then A is decidable in C. (Contributed by BJ, 19-Feb-2022.)

Hypotheses

Ref	Expression
decidin.ss	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
decidin.a	⊢ (𝜑 → 𝐴 DECIDin 𝐵)
decidin.b	⊢ (𝜑 → 𝐵 DECIDin 𝐶)

Assertion

Ref	Expression
decidin	⊢ (𝜑 → 𝐴 DECIDin 𝐶)

Proof of Theorem decidin

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		decidin.b	. . . 4 ⊢ (𝜑 → 𝐵 DECIDin 𝐶)
2		decidi 13676	. . . 4 ⊢ (𝐵 DECIDin 𝐶 → (𝑥 ∈ 𝐶 → (𝑥 ∈ 𝐵 ∨ ¬ 𝑥 ∈ 𝐵)))
3	1, 2	syl 14	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐶 → (𝑥 ∈ 𝐵 ∨ ¬ 𝑥 ∈ 𝐵)))
4		decidin.a	. . . . 5 ⊢ (𝜑 → 𝐴 DECIDin 𝐵)
5		decidi 13676	. . . . 5 ⊢ (𝐴 DECIDin 𝐵 → (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴)))
6	4, 5	syl 14	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴)))
7		decidin.ss	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
8	7	ssneld 3144	. . . . 5 ⊢ (𝜑 → (¬ 𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴))
9		olc 701	. . . . 5 ⊢ (¬ 𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴))
10	8, 9	syl6 33	. . . 4 ⊢ (𝜑 → (¬ 𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴)))
11	6, 10	jaod 707	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∨ ¬ 𝑥 ∈ 𝐵) → (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴)))
12	3, 11	syld 45	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐶 → (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴)))
13	12	decidr 13677	1 ⊢ (𝜑 → 𝐴 DECIDin 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 698   ∈ wcel 2136   ⊆ wss 3116   DECID_in wdcin 13674

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-dc 825  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-in 3122  df-ss 3129  df-dcin 13675

This theorem is referenced by:  sumdc2  13680