Theorem 1sdom2ALT

Description: Alternate proof of 1sdom2 , shorter but requiring ax-un . (Contributed by NM, 4-Apr-2007) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	1sdom2ALT	$$⊢1_{𝑜}\prec 2_{𝑜}$$

Proof

Step	Hyp	Ref	Expression
1		1onn	$$⊢1_{𝑜}\in \mathrm{\omega }$$
2		php4	$$⊢1_{𝑜}\in \mathrm{\omega }\to 1_{𝑜}\prec \mathrm{suc}{1}_{𝑜}$$
3	1 2	ax-mp	$$⊢1_{𝑜}\prec \mathrm{suc}{1}_{𝑜}$$
4		df-2o	$$⊢2_{𝑜}=\mathrm{suc}{1}_{𝑜}$$
5	3 4	breqtrri	$$⊢1_{𝑜}\prec 2_{𝑜}$$