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Mirrors > Home > ILE Home > Th. List > df-reap | GIF version |
Description: Define real apartness. Definition in Section 11.2.1 of [HoTT], p. (varies). Although #ℝ is an apartness relation on the reals (see df-ap 8488 for more discussion of apartness relations), for our purposes it is just a stepping stone to defining # which is an apartness relation on complex numbers. On the reals, #ℝ and # agree (apreap 8493). (Contributed by Jim Kingdon, 26-Jan-2020.) |
Ref | Expression |
---|---|
df-reap | ⊢ #ℝ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑥 < 𝑦 ∨ 𝑦 < 𝑥))} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | creap 8480 | . 2 class #ℝ | |
2 | vx | . . . . . . 7 setvar 𝑥 | |
3 | 2 | cv 1347 | . . . . . 6 class 𝑥 |
4 | cr 7760 | . . . . . 6 class ℝ | |
5 | 3, 4 | wcel 2141 | . . . . 5 wff 𝑥 ∈ ℝ |
6 | vy | . . . . . . 7 setvar 𝑦 | |
7 | 6 | cv 1347 | . . . . . 6 class 𝑦 |
8 | 7, 4 | wcel 2141 | . . . . 5 wff 𝑦 ∈ ℝ |
9 | 5, 8 | wa 103 | . . . 4 wff (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) |
10 | clt 7941 | . . . . . 6 class < | |
11 | 3, 7, 10 | wbr 3987 | . . . . 5 wff 𝑥 < 𝑦 |
12 | 7, 3, 10 | wbr 3987 | . . . . 5 wff 𝑦 < 𝑥 |
13 | 11, 12 | wo 703 | . . . 4 wff (𝑥 < 𝑦 ∨ 𝑦 < 𝑥) |
14 | 9, 13 | wa 103 | . . 3 wff ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑥 < 𝑦 ∨ 𝑦 < 𝑥)) |
15 | 14, 2, 6 | copab 4047 | . 2 class {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑥 < 𝑦 ∨ 𝑦 < 𝑥))} |
16 | 1, 15 | wceq 1348 | 1 wff #ℝ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑥 < 𝑦 ∨ 𝑦 < 𝑥))} |
Colors of variables: wff set class |
This definition is referenced by: reapval 8482 |
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