Cardinality of Subset Relation on Power Set of Finite Set

Theorem
Let $S$ be a set such that:
 * $\left|{S}\right| = n$

where $\left|{S}\right|$ denotes the cardinality of $S$.

From Subset Relation on Power Set is Partial Ordering we have that $\left({\mathcal P \left({S}\right), \subseteq}\right)$ is a poset.

The cardinality of $\subseteq$ is:


 * $\displaystyle \sum_{j=0}^n \sum_{i=0}^{n-j} \binom {n-j}{i}$