Theorem 2onn

Description: The ordinal 2 is a natural number. For a shorter proof using Peano's postulates that depends on ax-un , see 2onnALT . (Contributed by NM, 28-Sep-2004) Avoid ax-un . (Revised by BTernaryTau, 1-Dec-2024)

	Ref	Expression
Assertion	2onn	⊢ 2o ∈ ω

Proof

Step	Hyp	Ref	Expression
1		2on	⊢ 2o ∈ On
2		2ellim	⊢ ( Lim 𝑥 → 2o ∈ 𝑥 )
3	2	ax-gen	⊢ ∀ 𝑥 ( Lim 𝑥 → 2o ∈ 𝑥 )
4		elom	⊢ ( 2o ∈ ω ↔ ( 2o ∈ On ∧ ∀ 𝑥 ( Lim 𝑥 → 2o ∈ 𝑥 ) ) )
5	1 3 4	mpbir2an	⊢ 2o ∈ ω