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Metamath Proof Explorer

Theorem eluzelz2

Description: A member of an upper set of integers is an integer. (Contributed by Glauco Siliprandi, 23-Oct-2021)

		Ref	Expression
	Hypothesis	eluzelz2.1	$⊢ Z = ℤ_{\geq M}$
	Assertion	eluzelz2	$⊢ N \in Z \to N \in ℤ$

Step	Hyp	Ref	Expression
1		eluzelz2.1	$⊢ Z = ℤ_{\geq M}$
2	1	eleq2i	$⊢ N \in Z \leftrightarrow N \in ℤ_{\geq M}$
3	2	biimpi	$⊢ N \in Z \to N \in ℤ_{\geq M}$
4		eluzelz	$⊢ N \in ℤ_{\geq M} \to N \in ℤ$
5	3 4	syl	$⊢ N \in Z \to N \in ℤ$