Theorem uzssd

Description: Subset relationship for two sets of upper integers. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypothesis	uzssd.1	$$⊢\phi \to N\in {\mathbb{Z}}_{\ge M}$$
Assertion	uzssd	$$⊢\phi \to {\mathbb{Z}}_{\ge N}\subseteq {\mathbb{Z}}_{\ge M}$$

Proof

Step	Hyp	Ref	Expression
1		uzssd.1	$$⊢\phi \to N\in {\mathbb{Z}}_{\ge M}$$
2		uzss	$$⊢N\in {\mathbb{Z}}_{\ge M}\to {\mathbb{Z}}_{\ge N}\subseteq {\mathbb{Z}}_{\ge M}$$
3	1 2	syl	$$⊢\phi \to {\mathbb{Z}}_{\ge N}\subseteq {\mathbb{Z}}_{\ge M}$$