Theorem nnz

Description: A positive integer is an integer. (Contributed by NM, 9-May-2004) Reduce dependencies on axioms. (Revised by Steven Nguyen, 29-Nov-2022)

	Ref	Expression
Assertion	nnz	⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℤ )

Proof

Step	Hyp	Ref	Expression
1		nnre	⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℝ )
2		3mix2	⊢ ( 𝑁 ∈ ℕ → ( 𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ - 𝑁 ∈ ℕ ) )
3		elz	⊢ ( 𝑁 ∈ ℤ ↔ ( 𝑁 ∈ ℝ ∧ ( 𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ - 𝑁 ∈ ℕ ) ) )
4	1 2 3	sylanbrc	⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℤ )