Theorem 1onnALT

Description: Shorter proof of 1onn using Peano's postulates that depends on ax-un . (Contributed by NM, 29-Oct-1995) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	1onnALT	$$⊢1_{𝑜}\in \mathrm{\omega }$$

Proof

Step	Hyp	Ref	Expression
1		df-1o	$$⊢1_{𝑜}=\mathrm{suc}\varnothing$$
2		peano1	$$⊢\varnothing \in \mathrm{\omega }$$
3		peano2	$$⊢\varnothing \in \mathrm{\omega }\to \mathrm{suc}\varnothing \in \mathrm{\omega }$$
4	2 3	ax-mp	$$⊢\mathrm{suc}\varnothing \in \mathrm{\omega }$$
5	1 4	eqeltri	$$⊢1_{𝑜}\in \mathrm{\omega }$$