Theorem elong

Description: An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994)

	Ref	Expression
Assertion	elong	$$⊢A\in V\to \left(A\in \mathrm{On}\leftrightarrow \mathrm{Ord}A\right)$$

Proof

Step	Hyp	Ref	Expression
1		ordeq	$$⊢y=x\to \left(\mathrm{Ord}y\leftrightarrow \mathrm{Ord}x\right)$$
2		ordeq	$$⊢x=A\to \left(\mathrm{Ord}x\leftrightarrow \mathrm{Ord}A\right)$$
3		df-on	$$⊢\mathrm{On}=\left\{y|\mathrm{Ord}y\right\}$$
4	1 2 3	elab2gw	$$⊢A\in V\to \left(A\in \mathrm{On}\leftrightarrow \mathrm{Ord}A\right)$$