Prove that $A \cap B = A \Rightarrow B^{c} \subseteq A^{c}$

Question

Prove that $A \cap B = A \Rightarrow B^{c} \subseteq A^{c}$

My attempt:

Let $x\in B^{c}$. We want to show that $\forall x \in B^{c}, x\in A^{c}$

Since $x\in B^{c}, x\notin B$

And since $A \cap B = A$, This means $x \in A$ and $x \notin A^{c}$

Thus, $A \cap B = A \nRightarrow B^{c} \subseteq A^{c}$

Is this correct?

Answer 1

Continue like this; Since $x\in B^{c}, x\notin B$

and hence $x\notin A \cap B = A$, This means $x \notin A$ and therefore $x \in A^{c}.$