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Mirrors > Home > ILE Home > Th. List > df-uz | GIF version |
Description: Define a function whose value at 𝑗 is the semi-infinite set of contiguous integers starting at 𝑗, which we will also call the upper integers starting at 𝑗. Read "ℤ≥‘𝑀 " as "the set of integers greater than or equal to 𝑀." See uzval 9328 for its value, uzssz 9345 for its relationship to ℤ, nnuz 9361 and nn0uz 9360 for its relationships to ℕ and ℕ0, and eluz1 9330 and eluz2 9332 for its membership relations. (Contributed by NM, 5-Sep-2005.) |
Ref | Expression |
---|---|
df-uz | ⊢ ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cuz 9326 | . 2 class ℤ≥ | |
2 | vj | . . 3 setvar 𝑗 | |
3 | cz 9054 | . . 3 class ℤ | |
4 | 2 | cv 1330 | . . . . 5 class 𝑗 |
5 | vk | . . . . . 6 setvar 𝑘 | |
6 | 5 | cv 1330 | . . . . 5 class 𝑘 |
7 | cle 7801 | . . . . 5 class ≤ | |
8 | 4, 6, 7 | wbr 3929 | . . . 4 wff 𝑗 ≤ 𝑘 |
9 | 8, 5, 3 | crab 2420 | . . 3 class {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} |
10 | 2, 3, 9 | cmpt 3989 | . 2 class (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) |
11 | 1, 10 | wceq 1331 | 1 wff ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) |
Colors of variables: wff set class |
This definition is referenced by: uzval 9328 uzf 9329 uzennn 10209 |
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