Proving $A \cap \emptyset = \emptyset$

Question

Is this correct formal way to prove $A \cap \emptyset = \emptyset$ ?

$$ \begin{split} x \in A \cap \emptyset & \iff x \in \Big[x \ | \ (x \in A) \land (x \in \emptyset)\Big] \\ & \iff (x \in A) \land (x \in \emptyset) \\ & \iff (x \in A) \land F \\ & \iff F \\ & \iff x \in \emptyset \end{split}$$

Answer 1

Suppose that $A \cap \emptyset \ne \emptyset$, so it exists an $x$ that belongs to $A \cap \emptyset$. But then $x \in \emptyset$ absurd.