Metamath Proof Explorer

Theorem onelon

Description: An element of an ordinal number is an ordinal number. Theorem 2.2(iii) of BellMachover p. 469. Lemma 1.3 of Schloeder p. 1. (Contributed by NM, 26-Oct-2003)

		Ref	Expression
	Assertion	onelon	$⊢ (A \in On \land B \in A) \to B \in On$

Proof

Step	Hyp	Ref	Expression
1		eloni	$⊢ A \in On \to Ord A$
2		ordelon	$⊢ (Ord A \land B \in A) \to B \in On$
3	1 2	sylan	$⊢ (A \in On \land B \in A) \to B \in On$