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	$$⊢sup\left({\mathbb{N}}_0,ℝ,<\right)=0$$

Proof

Step	Hyp	Ref	Expression
1		nn0uz	$$⊢{\mathbb{N}}_0={\mathbb{Z}}_{\ge 0}$$
2	1	infeq1i	$$⊢sup\left({\mathbb{N}}_0,ℝ,<\right)=sup\left({\mathbb{Z}}_{\ge 0},ℝ,<\right)$$
3		0z	$$⊢0\in \mathbb{Z}$$
4	3	uzinfi	$$⊢sup\left({\mathbb{Z}}_{\ge 0},ℝ,<\right)=0$$
5	2 4	eqtri	$$⊢sup\left({\mathbb{N}}_0,ℝ,<\right)=0$$