Fix any errors in my proof (beginner)

Last time I did this I wasn't aware how difficult the proof was, but this one should be fairly straightforward. I upped the bounty regardless, though.

Critique my proof of the following theorem on correctness, structure, etc.

Theorem. Suppose $A$ and $B$ are sets. Prove that if $A \cap B = A$, then $A \subseteq B$.

Proof. Suppose $A \cap B = A$. Let $x$ be arbitrary and $x \in A$. Because $A \cap B = A$, it follows that $x \in A$ and $x \in B$. Because x is arbitrary, it must be the case that $\forall x(x\in A \implies x \in B)$, so $A \subseteq B$. Therefore, if $A\cap B = A$, then $A \subseteq B$.