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Metamath Proof Explorer

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	$⊢ 2_{𝑜} \in ω$

Proof

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