State and prove schroeder-bernstein theorem on cardinal numbers?

The Schröder-Bernstein theorem states that if there are injective functions ( f: A \to B ) and ( g: B \to A ) between two sets ( A ) and ( B ), then there exists a bijective function ( h: A \to B ), implying that the cardinalities of ( A ) and ( B ) are equal (denoted ( |A| = |B| )).

Proof: Construct a relation ( R ) where ( x R y ) if there exists a finite sequence of applications of ( f ) and ( g ) leading from ( x ) to ( y ). Using this relation, partition ( A ) and ( B ) into equivalence classes. The function ( h ) is defined to map each class in ( A ) to a unique representative in ( B ). This construction ensures that ( h ) is well-defined and bijective, thus proving ( |A| = |B| ).