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

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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.

$\blacksquare$


Sources