Cardinality of Subset of Finite Set

Theorem
Let $A$ and $B$ be finite sets such that $A \subseteq B$.

Let $\left\lvert{B}\right\rvert = n$.

Then $\left\lvert{A}\right\rvert \le n$.

Proof
Let $A \subseteq B$.

There are two cases:

$(1): \quad A \ne B$

In this case:
 * $A \subsetneqq B$

and from Cardinality of Proper Subset of Finite Set:
 * $\left\lvert{A}\right\rvert < n$

$(2): \quad A = B$

In this case:
 * $\left\lvert{A}\right\rvert = \left\lvert{B}\right\rvert$

and so:
 * $\left\lvert{A}\right\rvert = n$

In both cases:
 * $\left\lvert{A}\right\rvert \le n$

Hence the result.

Also see

 * Subset of Finite Set is Finite