Theorem ord3

Description: Ordinal 3 is an ordinal class. (Contributed by BTernaryTau, 6-Jan-2025)

	Ref	Expression
Assertion	ord3	$$⊢\mathrm{Ord}{3}_{𝑜}$$

Proof

Step	Hyp	Ref	Expression
1		2on	$$⊢2_{𝑜}\in \mathrm{On}$$
2		eloni	$$⊢2_{𝑜}\in \mathrm{On}\to \mathrm{Ord}{2}_{𝑜}$$
3		ordsuci	$$⊢\mathrm{Ord}{2}_{𝑜}\to \mathrm{Ord}\mathrm{suc}{2}_{𝑜}$$
4	1 2 3	mp2b	$$⊢\mathrm{Ord}\mathrm{suc}{2}_{𝑜}$$
5		df-3o	$$⊢3_{𝑜}=\mathrm{suc}{2}_{𝑜}$$
6		ordeq	$$⊢3_{𝑜}=\mathrm{suc}{2}_{𝑜}\to \left(\mathrm{Ord}{3}_{𝑜}\leftrightarrow \mathrm{Ord}\mathrm{suc}{2}_{𝑜}\right)$$
7	5 6	ax-mp	$$⊢\mathrm{Ord}{3}_{𝑜}\leftrightarrow \mathrm{Ord}\mathrm{suc}{2}_{𝑜}$$
8	4 7	mpbir	$$⊢\mathrm{Ord}{3}_{𝑜}$$