Proof by contradiction(sets)

Suppose the complement of the universal set is not empty.

Then the complement of the universal set has at least one element. Let $x$ be an element of the the complement.

But $x$ exists. so it is in the universal set.

So $x$ can not be in the complement of the universal set.

==== oops, old answer: I didn't realize you specifically wanted a prove by contradiction=====

It should follow directly by definitions.

If $U$ is the universal set, then $A^c = {xin U| xnot in A}={x| x in U,$ and $x in A}$.

So by definition $U^c = {xin U| xnot in U}={x| xin U, x not in U} = emptyset$.