Theorem elfzel2

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

	Ref	Expression
Assertion	elfzel2	$$⊢K\in \left(M\dots N\right)\to N\in \mathbb{Z}$$

Proof

Step	Hyp	Ref	Expression
1		elfzuz3	$$⊢K\in \left(M\dots N\right)\to N\in {\mathbb{Z}}_{\ge K}$$
2		eluzelz	$$⊢N\in {\mathbb{Z}}_{\ge K}\to N\in \mathbb{Z}$$
3	1 2	syl	$$⊢K\in \left(M\dots N\right)\to N\in \mathbb{Z}$$