Theorem decidin 16393

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 16391	. . . 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 16391	. . . . 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 3229	. . . . 5 ⊢ (𝜑 → (¬ 𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴))
9		olc 718	. . . . 5 ⊢ (¬ 𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴))
10	8, 9	syl6 33	. . . 4 ⊢ (𝜑 → (¬ 𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴)))
11	6, 10	jaod 724	. . 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 16392	1 ⊢ (𝜑 → 𝐴 DECIDin 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 715   ∈ wcel 2202   ⊆ wss 3200   DECID_in wdcin 16389

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-dc 842  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-in 3206  df-ss 3213  df-dcin 16390

This theorem is referenced by:  sumdc2  16395