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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 9519 for its value, uzssz 9536 for its relationship to ℤ, nnuz 9552 and nn0uz 9551 for its relationships to ℕ and ℕ0, and eluz1 9521 and eluz2 9523 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 9517 | . 2 class ℤ≥ | |
2 | vj | . . 3 setvar 𝑗 | |
3 | cz 9242 | . . 3 class ℤ | |
4 | 2 | cv 1352 | . . . . 5 class 𝑗 |
5 | vk | . . . . . 6 setvar 𝑘 | |
6 | 5 | cv 1352 | . . . . 5 class 𝑘 |
7 | cle 7983 | . . . . 5 class ≤ | |
8 | 4, 6, 7 | wbr 4000 | . . . 4 wff 𝑗 ≤ 𝑘 |
9 | 8, 5, 3 | crab 2459 | . . 3 class {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} |
10 | 2, 3, 9 | cmpt 4061 | . 2 class (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) |
11 | 1, 10 | wceq 1353 | 1 wff ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) |
Colors of variables: wff set class |
This definition is referenced by: uzval 9519 uzf 9520 uzennn 10422 |
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