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Mirrors > Home > ILE Home > Th. List > dcextest | GIF version |
Description: If it is decidable whether {𝑥 ∣ 𝜑} is a set, then ¬ 𝜑 is decidable (where 𝑥 does not occur in 𝜑). From this fact, we can deduce (outside the formal system, since we cannot quantify over classes) that if it is decidable whether any class is a set, then "weak excluded middle" (that is, any negated proposition ¬ 𝜑 is decidable) holds. (Contributed by Jim Kingdon, 3-Jul-2022.) |
Ref | Expression |
---|---|
dcextest.ex | ⊢ DECID {𝑥 ∣ 𝜑} ∈ V |
Ref | Expression |
---|---|
dcextest | ⊢ DECID ¬ 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dcextest.ex | . . . 4 ⊢ DECID {𝑥 ∣ 𝜑} ∈ V | |
2 | exmiddc 821 | . . . 4 ⊢ (DECID {𝑥 ∣ 𝜑} ∈ V → ({𝑥 ∣ 𝜑} ∈ V ∨ ¬ {𝑥 ∣ 𝜑} ∈ V)) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ ({𝑥 ∣ 𝜑} ∈ V ∨ ¬ {𝑥 ∣ 𝜑} ∈ V) |
4 | vprc 4060 | . . . . . . 7 ⊢ ¬ V ∈ V | |
5 | df-v 2688 | . . . . . . . . 9 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥} | |
6 | equid 1677 | . . . . . . . . . . 11 ⊢ 𝑥 = 𝑥 | |
7 | pm5.1im 172 | . . . . . . . . . . 11 ⊢ (𝑥 = 𝑥 → (𝜑 → (𝑥 = 𝑥 ↔ 𝜑))) | |
8 | 6, 7 | ax-mp 5 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑥 = 𝑥 ↔ 𝜑)) |
9 | 8 | abbidv 2257 | . . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ 𝜑}) |
10 | 5, 9 | syl5req 2185 | . . . . . . . 8 ⊢ (𝜑 → {𝑥 ∣ 𝜑} = V) |
11 | 10 | eleq1d 2208 | . . . . . . 7 ⊢ (𝜑 → ({𝑥 ∣ 𝜑} ∈ V ↔ V ∈ V)) |
12 | 4, 11 | mtbiri 664 | . . . . . 6 ⊢ (𝜑 → ¬ {𝑥 ∣ 𝜑} ∈ V) |
13 | 12 | con2i 616 | . . . . 5 ⊢ ({𝑥 ∣ 𝜑} ∈ V → ¬ 𝜑) |
14 | vex 2689 | . . . . . . . . . 10 ⊢ 𝑦 ∈ V | |
15 | biidd 171 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜑)) | |
16 | 14, 15 | elab 2828 | . . . . . . . . 9 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) |
17 | 16 | notbii 657 | . . . . . . . 8 ⊢ (¬ 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ¬ 𝜑) |
18 | 17 | biimpri 132 | . . . . . . 7 ⊢ (¬ 𝜑 → ¬ 𝑦 ∈ {𝑥 ∣ 𝜑}) |
19 | 18 | eq0rdv 3407 | . . . . . 6 ⊢ (¬ 𝜑 → {𝑥 ∣ 𝜑} = ∅) |
20 | 0ex 4055 | . . . . . 6 ⊢ ∅ ∈ V | |
21 | 19, 20 | eqeltrdi 2230 | . . . . 5 ⊢ (¬ 𝜑 → {𝑥 ∣ 𝜑} ∈ V) |
22 | 13, 21 | impbii 125 | . . . 4 ⊢ ({𝑥 ∣ 𝜑} ∈ V ↔ ¬ 𝜑) |
23 | 22 | notbii 657 | . . . 4 ⊢ (¬ {𝑥 ∣ 𝜑} ∈ V ↔ ¬ ¬ 𝜑) |
24 | 22, 23 | orbi12i 753 | . . 3 ⊢ (({𝑥 ∣ 𝜑} ∈ V ∨ ¬ {𝑥 ∣ 𝜑} ∈ V) ↔ (¬ 𝜑 ∨ ¬ ¬ 𝜑)) |
25 | 3, 24 | mpbi 144 | . 2 ⊢ (¬ 𝜑 ∨ ¬ ¬ 𝜑) |
26 | df-dc 820 | . 2 ⊢ (DECID ¬ 𝜑 ↔ (¬ 𝜑 ∨ ¬ ¬ 𝜑)) | |
27 | 25, 26 | mpbir 145 | 1 ⊢ DECID ¬ 𝜑 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 ∨ wo 697 DECID wdc 819 ∈ wcel 1480 {cab 2125 Vcvv 2686 ∅c0 3363 |
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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-dif 3073 df-in 3077 df-ss 3084 df-nul 3364 |
This theorem is referenced by: (None) |
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