Theorem decidin 15872

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 15870	. . . 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 15870	. . . . 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 3199	. . . . 5 ⊢ (𝜑 → (¬ 𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴))
9		olc 713	. . . . 5 ⊢ (¬ 𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴))
10	8, 9	syl6 33	. . . 4 ⊢ (𝜑 → (¬ 𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴)))
11	6, 10	jaod 719	. . 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 15871	1 ⊢ (𝜑 → 𝐴 DECIDin 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 710   ∈ wcel 2177   ⊆ wss 3170   DECID_in wdcin 15868

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-ral 2490  df-v 2775  df-in 3176  df-ss 3183  df-dcin 15869

This theorem is referenced by:  sumdc2  15874