# Cardinality of Subset of Finite Set

## Theorem

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

Let

$\card B = n$

where $\card {\, \cdot \,}$ denotes cardinality.

Then $\card A \le n$.

## Proof

Let $A \subseteq B$.

There are two cases:

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

In this case:

$A \subsetneqq B$
$\card A < n$

$(2): \quad A = B$

In this case:

$\card A = \card B$

and so:

$\card A = n$

In both cases:

$\card A \le n$

Hence the result.

$\blacksquare$