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

Theorem
Let $X, Y$ be sets such that
 * $\left\vert X \right\vert \le \left\vert Y \right\vert$

and
 * $Y$ is finite,

where $\left\vert X \right\vert$ denotes the cardinality of $X$.

Then $X$ is finite.

Proof
By Finite iff Cardinality Less than Aleph Zero:
 * $\left\vert{Y}\right\vert < \aleph_0$

Then
 * $\left\vert{X}\right\vert < \aleph_0$

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