Theorem elfzuz2

Description: Implication of membership in a finite set of sequential integers. (Contributed by NM, 20-Sep-2005) (Revised by Mario Carneiro, 28-Apr-2015)

	Ref	Expression
Assertion	elfzuz2	$$⊢K\in \left(M\dots N\right)\to N\in {\mathbb{Z}}_{\ge M}$$

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		eqid	$$⊢{\mathbb{Z}}_{\ge M}={\mathbb{Z}}_{\ge M}$$
3	2	uztrn2	$$⊢\left(K\in {\mathbb{Z}}_{\ge M}\wedge N\in {\mathbb{Z}}_{\ge K}\right)\to N\in {\mathbb{Z}}_{\ge M}$$
4	1 3	sylbi	$$⊢K\in \left(M\dots N\right)\to N\in {\mathbb{Z}}_{\ge M}$$