Let $\mathfrak{C}$ be a class. I want to show that $\bigcup \mathfrak{C} = \{x: \exists C \in \mathfrak{C}, x \in C\}$ is a set exactly when $\mathfrak{C}$ is a set.
If $\mathfrak{C}$ is a set, then using the Axiom of Unions, we have that $\bigcup \mathfrak{C}$ is a set.
But I am unsure about how to show the other direction. Any advice is appreciated.
Edit: Would it be valid to do the following? Or am I using some kind of circular logic here?
Let $\mathfrak{X} = \{ \mathfrak{C} : \bigcup \mathfrak{C} \text{ is a set} \}$. Then $\mathfrak{C} \in \mathfrak{X}$ whenever $\bigcup \mathfrak{C}$ is a set, i.e. $\mathfrak{C}$ is a set whenever $\bigcup \mathfrak{C}$ is a set.
If $X$ = $\bigcup C$ is a set, then $C\subseteq P(X)$, so $C$ is a set.