Suppose $\displaystyle A$ is finite and$\displaystyle B\subseteq A$. Prove that $\displaystyle |A\backslash B|=|A|-|B|$.

My attempt: I try to prove it by using induction on $\displaystyle B$

Base case: Suppose $\displaystyle B$ has 1 element, then

$\displaystyle |A\backslash B|=|A|-1=|A|-|B|$

I don't think my base case is 'rigorous' enough, it feels there is something missing...

For induction step:

Suppose for $\displaystyle B$ having n elements, and $\displaystyle B\subseteq A$,then $\displaystyle |A\backslash B|=|A|-|B|$.

Now let $\displaystyle B$ has n+1 elements, let $\displaystyle b\in B$ be an arbitrary element, then $\displaystyle B'=B\backslash \{ b\}$ has n elements, then

$\displaystyle |A\backslash B'|=|A|-|B'|$

But after that, I have no idea how to proceed at all. Please provide any hint necessary. Thanks in advance.