Metamath Proof Explorer


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 2 𝑜 ω

Proof

Step Hyp Ref Expression
1 2on 2 𝑜 On
2 2ellim Lim x 2 𝑜 x
3 2 ax-gen x Lim x 2 𝑜 x
4 elom 2 𝑜 ω 2 𝑜 On x Lim x 2 𝑜 x
5 1 3 4 mpbir2an 2 𝑜 ω