Let $A$ and $B$ each be a set and assume that $A \cup B \subseteq A \cap B$. Prove that $A = B$.
Proof: Let $x$ be an element of $A$. Therefore $x$ is an element of $A \cup B$. Since $A \cup B$ is a subset of $A \cap B$, $x$ is an element of $A \cap B$. Therefore, $x$ is an element of $B$ also. Therefore, $A$ is a subset of $B$. Using the same argument with $A$ and $B$ switched around we find that $B$ is a subset of $A$. Thus $A = B$.
This proof seems kind of simple. Could someone check it for me and let me know if it is correct or not?