Theorem nninf

Description: The infimum of the set of positive integers is one. (Contributed by NM, 16-Jun-2005) (Revised by AV, 5-Sep-2020)

	Ref	Expression
Assertion	nninf	$$⊢sup\left(\mathbb{N},ℝ,<\right)=1$$

Proof

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