Theorem elfzubelfz

Description: If there is a member in a finite set of sequential integers, the upper bound is also a member of this finite set of sequential integers. (Contributed by Alexander van der Vekens, 31-May-2018)

	Ref	Expression
Assertion	elfzubelfz	$$⊢K\in \left(M\dots N\right)\to N\in \left(M\dots N\right)$$

Proof

Step	Hyp	Ref	Expression
1		elfzuz2	$$⊢K\in \left(M\dots N\right)\to N\in {\mathbb{Z}}_{\ge M}$$
2		eluzfz2	$$⊢N\in {\mathbb{Z}}_{\ge M}\to N\in \left(M\dots N\right)$$
3	1 2	syl	$$⊢K\in \left(M\dots N\right)\to N\in \left(M\dots N\right)$$