Theorem 0elfz

Description: 0 is an element of a finite set of sequential nonnegative integers with a nonnegative integer as upper bound. (Contributed by AV, 6-Apr-2018)

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

Proof

Step	Hyp	Ref	Expression
1		0nn0	⊢ 0 ∈ ℕ0
2	1	a1i	⊢ ( 𝑁 ∈ ℕ0 → 0 ∈ ℕ0 )
3		id	⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0 )
4		nn0ge0	⊢ ( 𝑁 ∈ ℕ0 → 0 ≤ 𝑁 )
5		elfz2nn0	⊢ ( 0 ∈ ( 0 ... 𝑁 ) ↔ ( 0 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝑁 ) )
6	2 3 4 5	syl3anbrc	⊢ ( 𝑁 ∈ ℕ0 → 0 ∈ ( 0 ... 𝑁 ) )