User:Dfeuer/Set is Subset of Power Set of Union
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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.
$\blacksquare$