Theorem uzid2

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

	Ref	Expression
Assertion	uzid2	⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 𝑁 ) → 𝑀 ∈ ( ℤ≥ ‘ 𝑀 ) )

Proof

Step	Hyp	Ref	Expression
1		eluzelz	⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 𝑁 ) → 𝑀 ∈ ℤ )
2		uzid	⊢ ( 𝑀 ∈ ℤ → 𝑀 ∈ ( ℤ≥ ‘ 𝑀 ) )
3	1 2	syl	⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 𝑁 ) → 𝑀 ∈ ( ℤ≥ ‘ 𝑀 ) )