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