Description: The infimum of the set of nonnegative integers is zero. (Contributed by NM, 16-Jun-2005) (Revised by AV, 5-Sep-2020)
Ref | Expression | ||
---|---|---|---|
Assertion | nn0inf | ⊢ inf ( ℕ0 , ℝ , < ) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0uz | ⊢ ℕ0 = ( ℤ≥ ‘ 0 ) | |
2 | 1 | infeq1i | ⊢ inf ( ℕ0 , ℝ , < ) = inf ( ( ℤ≥ ‘ 0 ) , ℝ , < ) |
3 | 0z | ⊢ 0 ∈ ℤ | |
4 | 3 | uzinfi | ⊢ inf ( ( ℤ≥ ‘ 0 ) , ℝ , < ) = 0 |
5 | 2 4 | eqtri | ⊢ inf ( ℕ0 , ℝ , < ) = 0 |