Theorem eluzfz

Description: Membership in a finite set of sequential integers. (Contributed by NM, 4-Oct-2005) (Revised by Mario Carneiro, 28-Apr-2015)

	Ref	Expression
Assertion	eluzfz	$$⊢\left(K\in {\mathbb{Z}}_{\ge M}\wedge N\in {\mathbb{Z}}_{\ge K}\right)\to K\in \left(M\dots N\right)$$

Proof

Step	Hyp	Ref	Expression
1		elfzuzb	$$⊢K\in \left(M\dots N\right)\leftrightarrow \left(K\in {\mathbb{Z}}_{\ge M}\wedge N\in {\mathbb{Z}}_{\ge K}\right)$$
2	1	biimpri	$$⊢\left(K\in {\mathbb{Z}}_{\ge M}\wedge N\in {\mathbb{Z}}_{\ge K}\right)\to K\in \left(M\dots N\right)$$