Theorem elfznn0

Description: A member of a finite set of sequential nonnegative integers is a nonnegative integer. (Contributed by NM, 5-Aug-2005) (Revised by Mario Carneiro, 28-Apr-2015)

	Ref	Expression
Assertion	elfznn0	⊢ ( 𝐾 ∈ ( 0 ... 𝑁 ) → 𝐾 ∈ ℕ0 )

Proof

Step	Hyp	Ref	Expression
1		elfz2nn0	⊢ ( 𝐾 ∈ ( 0 ... 𝑁 ) ↔ ( 𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝐾 ≤ 𝑁 ) )
2	1	simp1bi	⊢ ( 𝐾 ∈ ( 0 ... 𝑁 ) → 𝐾 ∈ ℕ0 )