Theorem peano2n0s

Description: Peano postulate: the successor of a non-negative surreal integer is a non-negative surreal integer. (Contributed by Scott Fenton, 17-Mar-2025)

	Ref	Expression
Assertion	peano2n0s	$$⊢A\in {\mathbb{N}}_{\mathrm{0s}}\to A{+}_s{1}_s\in {\mathbb{N}}_{\mathrm{0s}}$$

Proof

Step	Hyp	Ref	Expression
1		df-n0s	$$⊢{\mathbb{N}}_{\mathrm{0s}}=\mathrm{rec}\left(\left(x\in \mathrm{V}⟼x{+}_s{1}_s\right),0_s\right)\left[\mathrm{\omega }\right]$$
2	1	a1i	$$⊢A\in {\mathbb{N}}_{\mathrm{0s}}\to {\mathbb{N}}_{\mathrm{0s}}=\mathrm{rec}\left(\left(x\in \mathrm{V}⟼x{+}_s{1}_s\right),0_s\right)\left[\mathrm{\omega }\right]$$
3		0no	$$⊢0_s\in \mathrm{No}$$
4	3	a1i	$$⊢A\in {\mathbb{N}}_{\mathrm{0s}}\to 0_s\in \mathrm{No}$$
5		id	$$⊢A\in {\mathbb{N}}_{\mathrm{0s}}\to A\in {\mathbb{N}}_{\mathrm{0s}}$$
6	2 4 5	noseqp1	$$⊢A\in {\mathbb{N}}_{\mathrm{0s}}\to A{+}_s{1}_s\in {\mathbb{N}}_{\mathrm{0s}}$$