Can someone explain this proof of: "If $X$ is a connected set and $X \subset Y \subset \overline X$, then $Y$ is connected"?

Question

If $Y$ is not connected, $Y$ is union of nonempty seperated set $A$,$B$. Since $X$ is connected, Then $X \subset A$ or $X \subset B$. If $X \subset A$ then $\overline X \subset \overline A$ But since $A$, $B$ is seperated, $\overline A \cap B$ is empty and $B \subset \overline A$, so $B = \varnothing$, which derive a contradiction.

From this StackExchange answer.

I follow up until "$B \subset \overline A$". Why is this true?

Answer 1

As $X \subset Y \subset \overline{X}\subset \overline{A}$ and $B \subset Y$ it follows that $B \subset \overline{A}$.