Theorem elfznn0

Description: A member of a finite set of sequential nonnegative integers is a nonnegative integer. (Contributed by NM, 5-Aug-2005) (Revised by Mario Carneiro, 28-Apr-2015)

	Ref	Expression
Assertion	elfznn0	$$⊢K\in \left(0\dots N\right)\to K\in {\mathbb{N}}_0$$

Proof

Step	Hyp	Ref	Expression
1		elfz2nn0	$$⊢K\in \left(0\dots N\right)\leftrightarrow \left(K\in {\mathbb{N}}_0\wedge N\in {\mathbb{N}}_0\wedge K\le N\right)$$
2	1	simp1bi	$$⊢K\in \left(0\dots N\right)\to K\in {\mathbb{N}}_0$$