How can I prove DP completeness?
The DP-hardness of SAT-UNSAT is a direct consequence of the fact that SAT is NP-hard.

Suppose that $L \in \mathrm{DP}$. Then there exist languages $A \in \mathrm{NP}$ and $B \in \mathrm{coNP}$ such that $L = A \cap B$. Since SAT is NP-complete, there exists a polytime reduction $f$ such that $x \in A$ iff $f(x) \in \mathrm{SAT}$. Since UNSAT is coNP-complete, there exists a polytime reduction $g$ such that $x \in B$ iff $g(x) \in \mathrm{coSAT}$. You can check that the polytime reduction $h(x) = (f(x),g(x))$ satisfies $x \in L$ iff $h(x) \in \mathrm{SAT\text{-}UNSAT}$. This shows that SAT-UNSAT is DP-hard.

To complete the proof that SAT-UNSAT is DP-complete, you have to show that SAT-UNSAT is in DP. Presumably you already know how to do that.

Note that DP contains both SAT and UNSAT, and in particular differs from NP unless NP=coNP.