Description: Nonnegative integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005)
Ref | Expression | ||
---|---|---|---|
Assertion | nn0uz | ⊢ ℕ_{0} = ( ℤ_{≥} ‘ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0zrab | ⊢ ℕ_{0} = { 𝑘 ∈ ℤ ∣ 0 ≤ 𝑘 } | |
2 | 0z | ⊢ 0 ∈ ℤ | |
3 | uzval | ⊢ ( 0 ∈ ℤ → ( ℤ_{≥} ‘ 0 ) = { 𝑘 ∈ ℤ ∣ 0 ≤ 𝑘 } ) | |
4 | 2 3 | ax-mp | ⊢ ( ℤ_{≥} ‘ 0 ) = { 𝑘 ∈ ℤ ∣ 0 ≤ 𝑘 } |
5 | 1 4 | eqtr4i | ⊢ ℕ_{0} = ( ℤ_{≥} ‘ 0 ) |