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