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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	$⊢ 1_{𝑜} \in ω$

Proof

Step	Hyp	Ref	Expression
1		1on	$⊢ 1_{𝑜} \in On$
2		1ellim	$⊢ Lim x \to 1_{𝑜} \in x$
3	2	ax-gen	$⊢ \forall x (Lim x \to 1_{𝑜} \in x)$
4		elom	$⊢ 1_{𝑜} \in ω \leftrightarrow (1_{𝑜} \in On \land \forall x (Lim x \to 1_{𝑜} \in x))$
5	1 3 4	mpbir2an	$⊢ 1_{𝑜} \in ω$