The problem is to prove that there is only one empty set.
Let A and B be empty sets,
A is a subset of B and B is a subset of A (by the definition that the empty set is a subset of every set)
So A=B (by definition)
By convention, all empty sets are equal. Therefore, there is only one empty set.
I'm fine with everything except the last line. I don't know if its my wording but something just doesn't feel right about it.