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Mirrors > Home > ILE Home > Th. List > Mathboxes > decidin | GIF version |
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.) |
Ref | Expression |
---|---|
decidin.ss | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
decidin.a | ⊢ (𝜑 → 𝐴 DECIDin 𝐵) |
decidin.b | ⊢ (𝜑 → 𝐵 DECIDin 𝐶) |
Ref | Expression |
---|---|
decidin | ⊢ (𝜑 → 𝐴 DECIDin 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | decidin.b | . . . 4 ⊢ (𝜑 → 𝐵 DECIDin 𝐶) | |
2 | decidi 12991 | . . . 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 12991 | . . . . 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 3094 | . . . . 5 ⊢ (𝜑 → (¬ 𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴)) |
9 | olc 700 | . . . . 5 ⊢ (¬ 𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴)) | |
10 | 8, 9 | syl6 33 | . . . 4 ⊢ (𝜑 → (¬ 𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴))) |
11 | 6, 10 | jaod 706 | . . 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 12992 | 1 ⊢ (𝜑 → 𝐴 DECIDin 𝐶) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 697 ∈ wcel 1480 ⊆ wss 3066 DECIDin wdcin 12989 |
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 2119 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-v 2683 df-in 3072 df-ss 3079 df-dcin 12990 |
This theorem is referenced by: sumdc2 12995 |
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