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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 8475 for its value, uzssz 8492 for its relationship to ℤ, nnuz 8508 and nn0uz 8507 for its relationships to ℕ and ℕ0, and eluz1 8477 and eluz2 8479 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 8473 | . 2 class ℤ≥ | |
2 | vj | . . 3 setvar 𝑗 | |
3 | cz 8245 | . . 3 class ℤ | |
4 | 2 | cv 1242 | . . . . 5 class 𝑗 |
5 | vk | . . . . . 6 setvar 𝑘 | |
6 | 5 | cv 1242 | . . . . 5 class 𝑘 |
7 | cle 7061 | . . . . 5 class ≤ | |
8 | 4, 6, 7 | wbr 3764 | . . . 4 wff 𝑗 ≤ 𝑘 |
9 | 8, 5, 3 | crab 2310 | . . 3 class {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} |
10 | 2, 3, 9 | cmpt 3818 | . 2 class (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) |
11 | 1, 10 | wceq 1243 | 1 wff ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) |
Colors of variables: wff set class |
This definition is referenced by: uzval 8475 uzf 8476 |
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