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Mirrors > Home > ILE Home > Th. List > abvor0dc | GIF version |
Description: The class builder of a decidable proposition not containing the abstraction variable is either the universal class or the empty set. (Contributed by Jim Kingdon, 1-Aug-2018.) |
Ref | Expression |
---|---|
abvor0dc | ⊢ (DECID 𝜑 → ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dc 777 | . 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑)) | |
2 | id 19 | . . . . 5 ⊢ (𝜑 → 𝜑) | |
3 | vex 2605 | . . . . . 6 ⊢ 𝑥 ∈ V | |
4 | 3 | a1i 9 | . . . . 5 ⊢ (𝜑 → 𝑥 ∈ V) |
5 | 2, 4 | 2thd 173 | . . . 4 ⊢ (𝜑 → (𝜑 ↔ 𝑥 ∈ V)) |
6 | 5 | abbi1dv 2199 | . . 3 ⊢ (𝜑 → {𝑥 ∣ 𝜑} = V) |
7 | id 19 | . . . . 5 ⊢ (¬ 𝜑 → ¬ 𝜑) | |
8 | noel 3262 | . . . . . 6 ⊢ ¬ 𝑥 ∈ ∅ | |
9 | 8 | a1i 9 | . . . . 5 ⊢ (¬ 𝜑 → ¬ 𝑥 ∈ ∅) |
10 | 7, 9 | 2falsed 651 | . . . 4 ⊢ (¬ 𝜑 → (𝜑 ↔ 𝑥 ∈ ∅)) |
11 | 10 | abbi1dv 2199 | . . 3 ⊢ (¬ 𝜑 → {𝑥 ∣ 𝜑} = ∅) |
12 | 6, 11 | orim12i 709 | . 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅)) |
13 | 1, 12 | sylbi 119 | 1 ⊢ (DECID 𝜑 → ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 662 DECID wdc 776 = wceq 1285 ∈ wcel 1434 {cab 2068 Vcvv 2602 ∅c0 3258 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-tru 1288 df-nf 1391 df-sb 1687 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-v 2604 df-dif 2976 df-nul 3259 |
This theorem is referenced by: (None) |
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