Theorem 2on

Description: Ordinal 2 is an ordinal number. (Contributed by NM, 18-Feb-2004) (Proof shortened by Andrew Salmon, 12-Aug-2011) Avoid ax-un . (Revised by BTernaryTau, 30-Nov-2024)

	Ref	Expression
Assertion	2on	⊢ 2o ∈ On

Proof

Step	Hyp	Ref	Expression
1		df-2o	⊢ 2o = suc 1o
2		1on	⊢ 1o ∈ On
3		2oex	⊢ 2o ∈ V
4	1 3	eqeltrri	⊢ suc 1o ∈ V
5		sucexeloni	⊢ ( ( 1o ∈ On ∧ suc 1o ∈ V ) → suc 1o ∈ On )
6	2 4 5	mp2an	⊢ suc 1o ∈ On
7	1 6	eqeltri	⊢ 2o ∈ On