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	$$⊢Z={\mathbb{Z}}_{\ge M}$$
Assertion	eluzelz2	$$⊢N\in Z\to N\in \mathbb{Z}$$

Proof

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