If $A\cap B^\complement=\emptyset$ then $A\cap B=A$ I need help with this excercise.
If $A\cap B^\complement=\emptyset$ then $A\cap B=A$

I try
$$A\cap B^\complement =\emptyset$$
$$(A\cap B^\complement)\cap B =\emptyset \cap B$$
$$A\cap (B^\complement\cap B )= B$$
$$A=B$$
Is this reasoning correct?

For every $x\in A$, you have $x\not \in B^c$.
$\implies$For every $x\in A$, you have $x\in B$.
$\implies$ $A\subseteq B$
$\implies A\cap B=A$
