Co-meager contains a perfect set

Question

Let $A\subset\mathbb R$ be a meager set and $I\subset\mathbb R$ be an open interval. So, $I\setminus A$ is co-meager in I. I know, $I\setminus A$ has ccardinality $\mathfrak c$ and dense in $I.$ I did not see why $I\setminus A$ contains a perfect set. I might miss some improtant proteries for co-meager set. Any help will be appreciated greatly.