Theorem elnns2

Description: A positive surreal integer is a non-negative surreal integer greater than zero. (Contributed by Scott Fenton, 15-Apr-2025)

	Ref	Expression
Assertion	elnns2	⊢ ( 𝐴 ∈ ℕs ↔ ( 𝐴 ∈ ℕ0s ∧ 0s <s 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		elnns	⊢ ( 𝐴 ∈ ℕs ↔ ( 𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s ) )
2		nesym	⊢ ( 𝐴 ≠ 0s ↔ ¬ 0s = 𝐴 )
3		n0sge0	⊢ ( 𝐴 ∈ ℕ0s → 0s ≤s 𝐴 )
4		0sno	⊢ 0s ∈ No
5		n0sno	⊢ ( 𝐴 ∈ ℕ0s → 𝐴 ∈ No )
6		sleloe	⊢ ( ( 0s ∈ No ∧ 𝐴 ∈ No ) → ( 0s ≤s 𝐴 ↔ ( 0s <s 𝐴 ∨ 0s = 𝐴 ) ) )
7	4 5 6	sylancr	⊢ ( 𝐴 ∈ ℕ0s → ( 0s ≤s 𝐴 ↔ ( 0s <s 𝐴 ∨ 0s = 𝐴 ) ) )
8	3 7	mpbid	⊢ ( 𝐴 ∈ ℕ0s → ( 0s <s 𝐴 ∨ 0s = 𝐴 ) )
9	8	orcomd	⊢ ( 𝐴 ∈ ℕ0s → ( 0s = 𝐴 ∨ 0s <s 𝐴 ) )
10	9	ord	⊢ ( 𝐴 ∈ ℕ0s → ( ¬ 0s = 𝐴 → 0s <s 𝐴 ) )
11	2 10	biimtrid	⊢ ( 𝐴 ∈ ℕ0s → ( 𝐴 ≠ 0s → 0s <s 𝐴 ) )
12		sgt0ne0	⊢ ( 0s <s 𝐴 → 𝐴 ≠ 0s )
13	11 12	impbid1	⊢ ( 𝐴 ∈ ℕ0s → ( 𝐴 ≠ 0s ↔ 0s <s 𝐴 ) )
14	13	pm5.32i	⊢ ( ( 𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s ) ↔ ( 𝐴 ∈ ℕ0s ∧ 0s <s 𝐴 ) )
15	1 14	bitri	⊢ ( 𝐴 ∈ ℕs ↔ ( 𝐴 ∈ ℕ0s ∧ 0s <s 𝐴 ) )