This hardly needs a proof. To find \(A \cup B\text{,}\) you take everything in \(A\) and throw in everything in \(B\text{.}\) Since there is no element in both sets already, you will have \(\card{A}\) things and add \(\card{B}\) new things to it. This is what adding does! Of course, we can easily extend this to any number of disjoint sets.
in-context