Theorem 0n0s

Description: Peano postulate: 0s is a non-negative surreal integer. (Contributed by Scott Fenton, 17-Mar-2025)

	Ref	Expression
Assertion	0n0s	$$⊢0_s\in {\mathbb{N}}_{\mathrm{0s}}$$

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	$$⊢\top \to {\mathbb{N}}_{\mathrm{0s}}=\mathrm{rec}\left(\left(x\in \mathrm{V}⟼x{+}_s{1}_s\right),0_s\right)\left[\mathrm{\omega }\right]$$
3		0no	$$⊢0_s\in \mathrm{No}$$
4	3	a1i	$$⊢\top \to 0_s\in \mathrm{No}$$
5	2 4	noseq0	$$⊢\top \to 0_s\in {\mathbb{N}}_{\mathrm{0s}}$$
6	5	mptru	$$⊢0_s\in {\mathbb{N}}_{\mathrm{0s}}$$