The question provides an answer, saying:
- $A \subset A \cup B$, then, $A \cap B \subset B$, then $A \subset B$
- $B \subset A \cup B$, then, $A \cap B \subset A$, then $B \subset A$
(1) and (2) $\Leftrightarrow A = B$
I know that when two sets are equal, they they're the subsets of each other. What I don't understand is where the OP has brought this answer from.