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Metamath Proof Explorer

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	$⊢ K \in (M \dots N) \to M \in ℤ$

Proof

Step	Hyp	Ref	Expression
1		elfzuz	$⊢ K \in (M \dots N) \to K \in ℤ_{\geq M}$
2		eluzel2	$⊢ K \in ℤ_{\geq M} \to M \in ℤ$
3	1 2	syl	$⊢ K \in (M \dots N) \to M \in ℤ$