Prove $\mathbb A$ is an algebra or not.

Question

let $\mathbb A$ be a collection of all borel sets in $\mathbb R$ which include $[0,1]$ such that,

$\mathbb A$= { $A\in \mathbb B: [0,1] \subset A $}

I need to prove whether $\mathbb A$ is an algebra or not.

Let $A= (a,b)$ such that $[0,1] \subset [a,b].$

I am going to prove that complement is not included in this set.

So $ \overline A = (-\infty ,a) \cup (b,\infty )$ , which does not include $[0,1]$

So $ \overline A$ does not include in $\mathbb A$. Hence $\mathbb A$ is not an algebra.

Is this the correct way of doing this or did i do a mistake ?