You cannot prove that unicorns do not exist. Should you believe in unicorns? I hope not... it is fundamentally absurd to argue that something must be proven NOT to exist. If anything, the burden of "proof" lies with the believer.
Consider this: proof makes belief worthless. If one could prove that God exists, then it wouldn't matter if someone did not believe in God because it would be a known fact (just like it doesn't matter if you do not believe that 2+2=4... it would still be 4). Yet believers are always looking for "proof" to convince the non-believers!