Theorem n0p1nns

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

	Ref	Expression
Assertion	n0p1nns	⊢ ( 𝐴 ∈ ℕ0s → ( 𝐴 +s 1s ) ∈ ℕs )

Proof

Step	Hyp	Ref	Expression
1		oveq1	⊢ ( 𝑥 = 0s → ( 𝑥 +s 1s ) = ( 0s +s 1s ) )
2	1	eleq1d	⊢ ( 𝑥 = 0s → ( ( 𝑥 +s 1s ) ∈ ℕs ↔ ( 0s +s 1s ) ∈ ℕs ) )
3		oveq1	⊢ ( 𝑥 = 𝑦 → ( 𝑥 +s 1s ) = ( 𝑦 +s 1s ) )
4	3	eleq1d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 +s 1s ) ∈ ℕs ↔ ( 𝑦 +s 1s ) ∈ ℕs ) )
5		oveq1	⊢ ( 𝑥 = ( 𝑦 +s 1s ) → ( 𝑥 +s 1s ) = ( ( 𝑦 +s 1s ) +s 1s ) )
6	5	eleq1d	⊢ ( 𝑥 = ( 𝑦 +s 1s ) → ( ( 𝑥 +s 1s ) ∈ ℕs ↔ ( ( 𝑦 +s 1s ) +s 1s ) ∈ ℕs ) )
7		oveq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 +s 1s ) = ( 𝐴 +s 1s ) )
8	7	eleq1d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 +s 1s ) ∈ ℕs ↔ ( 𝐴 +s 1s ) ∈ ℕs ) )
9		1sno	⊢ 1s ∈ No
10		addslid	⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
11	9 10	ax-mp	⊢ ( 0s +s 1s ) = 1s
12		1nns	⊢ 1s ∈ ℕs
13	11 12	eqeltri	⊢ ( 0s +s 1s ) ∈ ℕs
14		peano2nns	⊢ ( ( 𝑦 +s 1s ) ∈ ℕs → ( ( 𝑦 +s 1s ) +s 1s ) ∈ ℕs )
15	14	a1i	⊢ ( 𝑦 ∈ ℕ0s → ( ( 𝑦 +s 1s ) ∈ ℕs → ( ( 𝑦 +s 1s ) +s 1s ) ∈ ℕs ) )
16	2 4 6 8 13 15	n0sind	⊢ ( 𝐴 ∈ ℕ0s → ( 𝐴 +s 1s ) ∈ ℕs )