Theorem elfz2nn

Description: A member of a finite set of sequential integers starting at 2 is a positive integer. (Contributed by AV, 5-Apr-2026)

	Ref	Expression
Assertion	elfz2nn	⊢ ( 𝐾 ∈ ( 2 ... 𝑁 ) → 𝐾 ∈ ℕ )

Proof

Step	Hyp	Ref	Expression
1		2eluzge1	⊢ 2 ∈ ( ℤ≥ ‘ 1 )
2		fzss1	⊢ ( 2 ∈ ( ℤ≥ ‘ 1 ) → ( 2 ... 𝑁 ) ⊆ ( 1 ... 𝑁 ) )
3	1 2	ax-mp	⊢ ( 2 ... 𝑁 ) ⊆ ( 1 ... 𝑁 )
4	3	sseli	⊢ ( 𝐾 ∈ ( 2 ... 𝑁 ) → 𝐾 ∈ ( 1 ... 𝑁 ) )
5		elfznn	⊢ ( 𝐾 ∈ ( 1 ... 𝑁 ) → 𝐾 ∈ ℕ )
6	4 5	syl	⊢ ( 𝐾 ∈ ( 2 ... 𝑁 ) → 𝐾 ∈ ℕ )