Theorem uzidd2

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

	Ref	Expression
Hypotheses	uzidd2.1	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
	uzidd2.2	⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
Assertion	uzidd2	⊢ ( 𝜑 → 𝑀 ∈ 𝑍 )

Proof

Step	Hyp	Ref	Expression
1		uzidd2.1	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
2		uzidd2.2	⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
3	1	uzidd	⊢ ( 𝜑 → 𝑀 ∈ ( ℤ≥ ‘ 𝑀 ) )
4	3 2	eleqtrrdi	⊢ ( 𝜑 → 𝑀 ∈ 𝑍 )