Korach

In 17,17-24 Hashem tells Moshe to prove that Aharon is supposed to be the Kohen Gadol by taking a staff from each of the Nesi'im and writing the Nasi's name on the staff (for Levi they wrote Aharon's name) and put them near the Aron Kodesh. Sure enough the next morning only Shevet Levi's staff blossomed. How does that prove it has to be Aharon who's the Kohen Gadol? It could still be anyone from Levi. Last time, the problem came from a Levi, wouldn't it make sense to somehow prove it can't be any other Levi other than Aharon?