How to prove lAUBl=lAl+lBl-lAnBl via a counting arguement. A,B finite sets

Can we simply argue:In counting the elements of AUB, we count the elements in A,nA and the elements in B, nB. But in couting the elements of A and elements of B we have counted x elements twice since the sets intersect. Therefore, we subtract the duplicate, AnB. Then because A,B are finite sets, AnB and AUB are finite sets.

I feel like this isn't a very strong proof. Is there a more rigorous way of proving the result?