If not, you could have published a proof that it was not. I knew somebody whose PhD project was a proof that all objects X are in Y. After rather some time he

If not, you could have published a proof that it was not. I knew somebody whose PhD project was a proof that all objects X are in Y. After rather some time he found instead that there are no objects X. That proof was not considered PhD material, although it seems like it should have been.