Suppose that $X$ has $N \in \{ 1, 2, 3, \ldots \}$ elements. We can write
\begin{equation}
\text{Part} X = \bigcup_{n = 0}^N \{ P \subseteq \mathcal{P}(X) : P \text{ partitions } X \text{ and } |P| = n \}
\end{equation}
The sets in the union are disjoint since each set consists of partitions of different sizes. Hence, applying result
R1838: Cardinality of finite set union of finite sets yields
\begin{equation}
\begin{split}
|\text{Part} X| & = \left| \bigcup_{n = 0}^N \{ P : P \text{ partitions } X \text{ and } |P| = n \} \right| \\
& = \sum_{n = 0}^N \left| \{ P : P \text{ partitions } X \text{ and } |P| = n \} \right| \\
& = \sum_{n = 0}^N {N \brace n}
\end{split}
\end{equation}
This is the desired result. $\square$
