Theorem decidin 13832

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 13830	. . . 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 13830	. . . . 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 3149	. . . . 5 ⊢ (𝜑 → (¬ 𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴))
9		olc 706	. . . . 5 ⊢ (¬ 𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴))
10	8, 9	syl6 33	. . . 4 ⊢ (𝜑 → (¬ 𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴)))
11	6, 10	jaod 712	. . 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 13831	1 ⊢ (𝜑 → 𝐴 DECIDin 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 703   ∈ wcel 2141   ⊆ wss 3121   DECID_in wdcin 13828

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-dc 830  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-in 3127  df-ss 3134  df-dcin 13829

This theorem is referenced by:  sumdc2  13834