Theorem nn0inf

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

Proof

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