Theorem nnssz

Description: Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002) Reduce dependencies on axioms. (Revised by Steven Nguyen, 29-Nov-2022)

	Ref	Expression
Assertion	nnssz	⊢ ℕ ⊆ ℤ

Proof

Step	Hyp	Ref	Expression
1		nnre	⊢ ( 𝑥 ∈ ℕ → 𝑥 ∈ ℝ )
2		3mix2	⊢ ( 𝑥 ∈ ℕ → ( 𝑥 = 0 ∨ 𝑥 ∈ ℕ ∨ - 𝑥 ∈ ℕ ) )
3		elz	⊢ ( 𝑥 ∈ ℤ ↔ ( 𝑥 ∈ ℝ ∧ ( 𝑥 = 0 ∨ 𝑥 ∈ ℕ ∨ - 𝑥 ∈ ℕ ) ) )
4	1 2 3	sylanbrc	⊢ ( 𝑥 ∈ ℕ → 𝑥 ∈ ℤ )
5	4	ssriv	⊢ ℕ ⊆ ℤ