Theorem elfzel1

Description: Membership in a finite set of sequential integer implies the lower bound is an integer. (Contributed by NM, 6-Sep-2005) (Revised by Mario Carneiro, 28-Apr-2015)

	Ref	Expression
Assertion	elfzel1	$$⊢K\in \left(M\dots N\right)\to M\in \mathbb{Z}$$

Proof

Step	Hyp	Ref	Expression
1		elfzuz	$$⊢K\in \left(M\dots N\right)\to K\in {\mathbb{Z}}_{\ge M}$$
2		eluzel2	$$⊢K\in {\mathbb{Z}}_{\ge M}\to M\in \mathbb{Z}$$
3	1 2	syl	$$⊢K\in \left(M\dots N\right)\to M\in \mathbb{Z}$$