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	⊢ 1o ∈ ω

Proof

Step	Hyp	Ref	Expression
1		df-1o	⊢ 1o = suc ∅
2		peano1	⊢ ∅ ∈ ω
3		peano2	⊢ ( ∅ ∈ ω → suc ∅ ∈ ω )
4	2 3	ax-mp	⊢ suc ∅ ∈ ω
5	1 4	eqeltri	⊢ 1o ∈ ω