First, don’t be concerned about the statements on cardinality.
Once you have the basic idea down, all the rest is easy.
Secondly, the above is not the Schroeder-Bernstein. But nonetheless it is easy to prove anything about finite sets.
The basic idea of a finite set is: Any finite set can be listed using positive integers as subscripts.
Thus if then we can write .
If we define by the result follows easily.