Is it possible to prove that something does not exist? I'm asking because I find it very difficult to think of any such idea.
Interesting question. I may be misunderstanding it but here's one possible answer.
In the second-century CE the Buddhist sage Nagarjuna logically proved that nothing really exists. Here the word 'really' would mean 'metaphysically'. Nobody has ever invalidated, falsified or refuted his result. If they ever do they will have falsified the Perennial philosophy, which claims the Unity of All. We find the same argument made in Bradley's Appearance and Reality and in 'Laws of Form by G.S Brown.
For a theory in which the Universe is a Unity it is necessary that 'things' do not exist independently or truly and that they are reducible. So yes, lots of people claim that nothing really exists and a few have proved it. Usually it is considered a truth only knowable in experience but logic supports it. This would be why nobody can prove that anything exists, and why existence (as we usually think of it) is paradoxical.
It is a difficult idea, but not incomprehensible with a bit of work.