Theorem eluz

Description: Membership in an upper set of integers. (Contributed by NM, 2-Oct-2005)

	Ref	Expression
Assertion	eluz	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ↔ 𝑀 ≤ 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		eluz1	⊢ ( 𝑀 ∈ ℤ → ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ↔ ( 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ) ) )
2	1	baibd	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ↔ 𝑀 ≤ 𝑁 ) )