Metamath Proof Explorer

Theorem eluzelz

Description: A member of an upper set of integers is an integer. (Contributed by NM, 6-Sep-2005)

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

Proof

Step Hyp Ref Expression
1 eluz2 ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ↔ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ) )
2 1 simp2bi ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑁 ∈ ℤ )