Please I need help proving the following question:

Given an infinite set A and a countable set B, prove that $\displaystyle {A}\cup{B}\sim{A}$.

I already proved the case when $\displaystyle {A}\cap{B}=\emptyset$ and when $\displaystyle {B}\subset{A}$.

I need help proving what happens when $\displaystyle {B}\nsubseteq{A}$ and $\displaystyle {A}\cap{B}\neq\emptyset$

10x in advance...