Theorem n0sex

Description: The set of all non-negative surreal integers exists. (Contributed by Scott Fenton, 17-Mar-2025)

	Ref	Expression
Assertion	n0sex	$$⊢{\mathbb{N}}_{\mathrm{0s}}\in \mathrm{V}$$

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	2	noseqex	$$⊢\top \to {\mathbb{N}}_{\mathrm{0s}}\in \mathrm{V}$$
4	3	mptru	$$⊢{\mathbb{N}}_{\mathrm{0s}}\in \mathrm{V}$$