Theorem uzssd2

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

	Ref	Expression
Hypotheses	uzssd2.1	⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
	uzssd2.2	⊢ ( 𝜑 → 𝑁 ∈ 𝑍 )
Assertion	uzssd2	⊢ ( 𝜑 → ( ℤ≥ ‘ 𝑁 ) ⊆ 𝑍 )

Proof

Step	Hyp	Ref	Expression
1		uzssd2.1	⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
2		uzssd2.2	⊢ ( 𝜑 → 𝑁 ∈ 𝑍 )
3	2 1	eleqtrdi	⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
4	3	uzssd	⊢ ( 𝜑 → ( ℤ≥ ‘ 𝑁 ) ⊆ ( ℤ≥ ‘ 𝑀 ) )
5	4 1	sseqtrrdi	⊢ ( 𝜑 → ( ℤ≥ ‘ 𝑁 ) ⊆ 𝑍 )