The rank of $y=\{\omega\}$ is infinite, but there is no $x$ whose rank is finite which is in $y$. What true is the converse, if $x\in y$ then the rank of $x$ is strictly less than the rank of $y$.
Again, recall the definition of rank. The rank of $y$ is the least ordinal larger than all the ranks of the elements of $y$. Your question, if true, would imply that any set which has an infinite rank would be infinite. In particular a singleton whose element is a set of infinite rank. This is obviously false, then.