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Metamath Proof Explorer

Theorem eluzel2

Description: Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005) (Revised by Mario Carneiro, 3-Nov-2013)

		Ref	Expression
	Assertion	eluzel2	$⊢ N \in ℤ_{\geq M} \to M \in ℤ$

Step	Hyp	Ref	Expression
1		elfvdm	$⊢ N \in ℤ_{\geq M} \to M \in dom ℤ_{\geq}$
2		uzf	$⊢ ℤ_{\geq} : ℤ ⟶ 𝒫 ℤ$
3	2	fdmi	$⊢ dom ℤ_{\geq} = ℤ$
4	1 3	eleqtrdi	$⊢ N \in ℤ_{\geq M} \to M \in ℤ$