Theorem 2onnALT

Description: Shorter proof of 2onn using Peano's postulates that depends on ax-un . (Contributed by NM, 28-Sep-2004) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	2onnALT	$$⊢2_{𝑜}\in \mathrm{\omega }$$

Proof

Step	Hyp	Ref	Expression
1		df-2o	$$⊢2_{𝑜}=\mathrm{suc}{1}_{𝑜}$$
2		1onn	$$⊢1_{𝑜}\in \mathrm{\omega }$$
3		peano2	$$⊢1_{𝑜}\in \mathrm{\omega }\to \mathrm{suc}{1}_{𝑜}\in \mathrm{\omega }$$
4	2 3	ax-mp	$$⊢\mathrm{suc}{1}_{𝑜}\in \mathrm{\omega }$$
5	1 4	eqeltri	$$⊢2_{𝑜}\in \mathrm{\omega }$$