Theorem n0zs

Description: A non-negative surreal integer is a surreal integer. (Contributed by Scott Fenton, 26-May-2025)

	Ref	Expression
Assertion	n0zs	⊢ ( 𝐴 ∈ ℕ0s → 𝐴 ∈ ℤs )

Proof

Step	Hyp	Ref	Expression
1		eln0s	⊢ ( 𝐴 ∈ ℕ0s ↔ ( 𝐴 ∈ ℕs ∨ 𝐴 = 0s ) )
2		nnzs	⊢ ( 𝐴 ∈ ℕs → 𝐴 ∈ ℤs )
3		id	⊢ ( 𝐴 = 0s → 𝐴 = 0s )
4		0zs	⊢ 0s ∈ ℤs
5	3 4	eqeltrdi	⊢ ( 𝐴 = 0s → 𝐴 ∈ ℤs )
6	2 5	jaoi	⊢ ( ( 𝐴 ∈ ℕs ∨ 𝐴 = 0s ) → 𝐴 ∈ ℤs )
7	1 6	sylbi	⊢ ( 𝐴 ∈ ℕ0s → 𝐴 ∈ ℤs )