elementary set theory - Does there exist a bijection between sets $Asetminus B$ and $A$?

Let $A$ and $B$ non-empty sets, A is infinite and B is countably infinite($\sim \mathbb{N}$). Prove that if A is not countably infinite and $B\subseteq A$, then exists a bijection between $A\setminus B$ and $A$.

I thought that i can use Schröder–Bernstein theorem, so i defined two injective function: $id:A\setminus B \rightarrow A $ and $id:A\rightarrow A\setminus B$ ($id$ identity function) Is that conclude that there is a a bijection between $A\setminus B$ and $A$? Thanks in advance.