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Metamath Proof Explorer

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	$⊢ K \in (M \dots N) \to N \in ℤ_{\geq M}$

Proof

Step	Hyp	Ref	Expression
1		elfzuzb	$⊢ K \in (M \dots N) \leftrightarrow (K \in ℤ_{\geq M} \land N \in ℤ_{\geq K})$
2		eqid	$⊢ ℤ_{\geq M} = ℤ_{\geq M}$
3	2	uztrn2	$⊢ (K \in ℤ_{\geq M} \land N \in ℤ_{\geq K}) \to N \in ℤ_{\geq M}$
4	1 3	sylbi	$⊢ K \in (M \dots N) \to N \in ℤ_{\geq M}$