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Metamath Proof Explorer

Theorem 1onn

Description: The ordinal 1 is a natural number. For a shorter proof using Peano's postulates that depends on ax-un , see 1onnALT . Lemma 2.2 of Schloeder p. 4. (Contributed by NM, 29-Oct-1995) Avoid ax-un . (Revised by BTernaryTau, 1-Dec-2024)

Ref Expression
Assertion 1onn 1o ∈ ω

Proof

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