User:Dfeuer/Set is Subset of Power Set of Union

Theorem
Let $a$ be a set.

Then $a \subseteq \mathcal P \left({\bigcup a}\right)$.

Proof
Let $x \in a$.

Then by User:Dfeuer/Element of Set is Subset of Union: $x \subseteq \bigcup a$.

Thus by the definition of power set, $x \in \mathcal P \left({\bigcup a}\right)$.

As this holds for all $x \in a$, the theorem holds.