Help with set proof: $A \cap B = A $ if and only if $A \subseteq B $. $A \cap B = A $ if and only if $A \subseteq B $


It's been a while since I've done this sort of proof. I can't think of how I would prove this statement. I'm too used to numerical proofs.

What kind of proof would I use and how?
Suppose $A\not\subset B$, then there exists an $a\in A$ such that $a\notin B$. Hence $a\notin A\cap B$. Thus $A\cap B\neq A$.

Conversely, if $A\cap B=A$ then $A\subset A\cap B$, so for each $a\in A$ we have that $a\in A\cap B$, which implies that $a\in B$. Thus $A\subset B$.
Let $A\cap B=A$ Show that $A\subseteq B$.

Let $A\subseteq B$, show that $A\cap B=A$.

Where is your problem?