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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