Theorem eluzelz2

Description: A member of an upper set of integers is an integer. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypothesis	eluzelz2.1	⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
Assertion	eluzelz2	⊢ ( 𝑁 ∈ 𝑍 → 𝑁 ∈ ℤ )

Proof

Step	Hyp	Ref	Expression
1		eluzelz2.1	⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
2	1	eleq2i	⊢ ( 𝑁 ∈ 𝑍 ↔ 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
3	2	biimpi	⊢ ( 𝑁 ∈ 𝑍 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
4		eluzelz	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑁 ∈ ℤ )
5	3 4	syl	⊢ ( 𝑁 ∈ 𝑍 → 𝑁 ∈ ℤ )