Theorem n0ssno

Description: The non-negative surreal integers are a subset of the surreals. (Contributed by Scott Fenton, 17-Mar-2025)

	Ref	Expression
Assertion	n0ssno	$$⊢{\mathbb{N}}_{\mathrm{0s}}\subseteq \mathrm{No}$$

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		0sno	$$⊢0_s\in \mathrm{No}$$
4	3	a1i	$$⊢\top \to 0_s\in \mathrm{No}$$
5	2 4	noseqssno	$$⊢\top \to {\mathbb{N}}_{\mathrm{0s}}\subseteq \mathrm{No}$$
6	5	mptru	$$⊢{\mathbb{N}}_{\mathrm{0s}}\subseteq \mathrm{No}$$