Theorem onnoxp

Description: Every ordinal maps to a surreal number. (Contributed by RP, 21-Sep-2023)

	Ref	Expression
Assertion	onnoxp	$$⊢A\in \mathrm{On}\to A\times \left\{2_{𝑜}\right\}\in \mathrm{No}$$

Proof

Step	Hyp	Ref	Expression
1		2oex	$$⊢2_{𝑜}\in \mathrm{V}$$
2	1	prid2	$$⊢2_{𝑜}\in \left\{1_{𝑜},2_{𝑜}\right\}$$
3		onnoxpg	$$⊢\left(A\in \mathrm{On}\wedge 2_{𝑜}\in \left\{1_{𝑜},2_{𝑜}\right\}\right)\to A\times \left\{2_{𝑜}\right\}\in \mathrm{No}$$
4	2 3	mpan2	$$⊢A\in \mathrm{On}\to A\times \left\{2_{𝑜}\right\}\in \mathrm{No}$$