Theorem limelon

Description: A limit ordinal class that is also a set is an ordinal number. (Contributed by NM, 26-Apr-2004)

	Ref	Expression
Assertion	limelon	$$⊢\left(A\in B\wedge \mathrm{Lim}A\right)\to A\in \mathrm{On}$$

Proof

Step	Hyp	Ref	Expression
1		limord	$$⊢\mathrm{Lim}A\to \mathrm{Ord}A$$
2		elong	$$⊢A\in B\to \left(A\in \mathrm{On}\leftrightarrow \mathrm{Ord}A\right)$$
3	1 2	syl5ibr	$$⊢A\in B\to \left(\mathrm{Lim}A\to A\in \mathrm{On}\right)$$
4	3	imp	$$⊢\left(A\in B\wedge \mathrm{Lim}A\right)\to A\in \mathrm{On}$$