Theorem elfzonn0

Description: A member of a half-open range of nonnegative integers is a nonnegative integer. (Contributed by Alexander van der Vekens, 21-May-2018)

	Ref	Expression
Assertion	elfzonn0	⊢ ( 𝐾 ∈ ( 0 ..^ 𝑁 ) → 𝐾 ∈ ℕ0 )

Proof

Step	Hyp	Ref	Expression
1		elfzouz	⊢ ( 𝐾 ∈ ( 0 ..^ 𝑁 ) → 𝐾 ∈ ( ℤ≥ ‘ 0 ) )
2		elnn0uz	⊢ ( 𝐾 ∈ ℕ0 ↔ 𝐾 ∈ ( ℤ≥ ‘ 0 ) )
3	1 2	sylibr	⊢ ( 𝐾 ∈ ( 0 ..^ 𝑁 ) → 𝐾 ∈ ℕ0 )