Fix any errors in my proof (beginnner)

Fix any errors in correctness, structure, etc.

Theorem. Suppose $B$ is a set and $F$ is a family of sets. If  $\bigcup F \subseteq B$, then $F \subseteq \wp(B)$.

Proof. Suppose $\bigcup F \subseteq B$. Let $y$ be an arbitrary element such that $y \in F$. Let x be an arbitrary element such that $x \in y$. It follows that $x \in \bigcup F$ and that $x \in B$. Because $\forall x(x \in y \implies x \in B)$, it follows that $y \subseteq B$ and by definition of powerset $y \in \wp(B)$. Because $\forall y(y \in F \implies y \in \wp (B)$), we can conclude that $F \subseteq \wp(B)$.