Most of the problems in my textbook have numeric solutions in the back of the book except the proofs. Is this proof correct?

Prove that if $A \cup B$ and $A \cap B$ are independent events, then either $P(A \cap B)=0$ or $P(A \cup B) = 0)$.

1. $P((A \cup B) \cap (A \cap B)) = P(A \cup B)P(A \cap B)$ (definition of independence)
2. $P(A \cap B)=P(A \cup B)P(A \cap B)$ (properties of sets and intersections)
3. Either $P(AB)=0$ or $P(A\cup B)=1$. In the first case we already proved the proposition. In the second case $P(A\cup B)=1\implies P(\bar{A} \cap \bar{B})=0$. $\square$