Set of Cardinality not Greater than Cardinality of Finite Set is Finite

Theorem
Let $X, Y$ be sets such that
 * $\card X \le \card Y$

and
 * $Y$ is finite,

where $\card X$ denotes the cardinality of $X$.

Then $X$ is finite.

Proof
By Finite iff Cardinality Less than Aleph Zero:
 * $\card Y < \aleph_0$

Then
 * $\card X < \aleph_0$

Thus by Finite iff Cardinality Less than Aleph Zero:
 * $X$ is a finite set.