Theorem eluzd

Description: Membership in an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypotheses	eluzd.1	⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
	eluzd.2	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
	eluzd.3	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
	eluzd.4	⊢ ( 𝜑 → 𝑀 ≤ 𝑁 )
Assertion	eluzd	⊢ ( 𝜑 → 𝑁 ∈ 𝑍 )

Proof

Step	Hyp	Ref	Expression
1		eluzd.1	⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
2		eluzd.2	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
3		eluzd.3	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
4		eluzd.4	⊢ ( 𝜑 → 𝑀 ≤ 𝑁 )
5		eluz2	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ↔ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ) )
6	2 3 4 5	syl3anbrc	⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
7	6 1	eleqtrrdi	⊢ ( 𝜑 → 𝑁 ∈ 𝑍 )