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Mirrors > Home > ILE Home > Th. List > Mathboxes > decidi | GIF version |
Description: Property of being decidable in another class. (Contributed by BJ, 19-Feb-2022.) |
Ref | Expression |
---|---|
decidi | ⊢ (𝐴 DECIDin 𝐵 → (𝑋 ∈ 𝐵 → (𝑋 ∈ 𝐴 ∨ ¬ 𝑋 ∈ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dcin 14168 | . 2 ⊢ (𝐴 DECIDin 𝐵 ↔ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴) | |
2 | df-dc 835 | . . . 4 ⊢ (DECID 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴)) | |
3 | 2 | ralbii 2483 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴)) |
4 | eleq1 2240 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴)) | |
5 | 4 | notbid 667 | . . . . 5 ⊢ (𝑥 = 𝑋 → (¬ 𝑥 ∈ 𝐴 ↔ ¬ 𝑋 ∈ 𝐴)) |
6 | 4, 5 | orbi12d 793 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴) ↔ (𝑋 ∈ 𝐴 ∨ ¬ 𝑋 ∈ 𝐴))) |
7 | 6 | rspccv 2838 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴) → (𝑋 ∈ 𝐵 → (𝑋 ∈ 𝐴 ∨ ¬ 𝑋 ∈ 𝐴))) |
8 | 3, 7 | sylbi 121 | . 2 ⊢ (∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴 → (𝑋 ∈ 𝐵 → (𝑋 ∈ 𝐴 ∨ ¬ 𝑋 ∈ 𝐴))) |
9 | 1, 8 | sylbi 121 | 1 ⊢ (𝐴 DECIDin 𝐵 → (𝑋 ∈ 𝐵 → (𝑋 ∈ 𝐴 ∨ ¬ 𝑋 ∈ 𝐴))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 708 DECID wdc 834 = wceq 1353 ∈ wcel 2148 ∀wral 2455 DECIDin wdcin 14167 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-v 2739 df-dcin 14168 |
This theorem is referenced by: decidin 14171 |
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