Uniqueness for Set in Family of sets This is an exercise in How to prove it by Velleman.
Suppose $\mathcal{F}$ is a family of sets. Prove that there is a unique set $A$ that has the following two properties:
(a) $\mathcal{F} \subseteq \mathcal{P}(A)$
(b) $\forall B(\mathcal{F} \subseteq \mathcal{P}(B) \rightarrow A \subseteq B)$
My approach so far: 
This set is obviously $A = \cup \mathcal{F}$. I have proven the existence part of the proof, but I am struggling with the uniqueness. 
My approach so far for proving uniqueness:
Let $P(X)$ denote $\mathcal{F} \subseteq \mathcal{P}(X) \land \forall B(\mathcal{F} \subseteq \mathcal{P}(B) \rightarrow A \subseteq B)$
My first approach was:
$\forall Y \forall Z( (P(Y) \land P(Z)) \rightarrow (Y=Z))$. I tried to somehow prove through $P(Y)$ that $Y = \cup \mathcal{F}$ and through $P(Z)$ that $Z = \cup \mathcal{F}$ and thus that $Y=Z$.
My second approach was:
Prove that $\forall X (P(X) \rightarrow X = A)$. This has also brought me nowhere so far. I tried proving the contra-positive, but got stuck in the case where I had to prove $\lnot P(X)$ when $X \nsubseteq A$.
I would be really grateful for all hints!

HINT: Let $A=\bigcup\mathscr{F}$, and suppose that $B$ also has the desired properties. Then on the one hand (b) applied to $A$ tells you that $A\subseteq B$, and on the other hand (b) applied to ... ?
