Sorry for this late entry, but I was on holiday until 21 Nov and have only just seen the question. The statement only needs proof if reality and existence are assumed to be different states of being. (Clearly the statements "Nothing unreal is real" or "Nothing non-existent exists" do NOT need proof, as they are tautological.) However, if you define "real" and "exists" so that they are not synonomous, I suspect that you would not think the issue worth pursuing. Eg suppose "real" is defined as only applying to objects, persons etc capable of being detected by physical methods; but "exists" as including both those real objects and abstract concepts, fictional persons etc. It would then be more or less self-evident that your original statment is untrue, as many "unreal" things (eg beauty, Oliver Twist) would "exist". However, if you exchanged the above definitions, so that "real" had the wider definition, and "exists" the narrower, then just as self-evidently your statment would be true, as nothing excluded from the wider class of "real" could be within the narrower class of "exists". However, as I said earlier, I suspect that you do not see any difference between "real" and "exists", in which case seeking a proof of the statement "Nothing unreal exists" is like seeking a proof of the statement "No number other than 2 is 2".