It looks like your question got cut off at the end. Could you clarify the full statement? Are you asking for a proof regarding a unique set $A \in P(C)$ (where $P(C)$ is the power set of $C$) that satisfies a certain property?

Once you clarify, I can help with the proof!

Let me know if you'd like further explanation on power sets, uniqueness proofs, or set theory in general.

Follow-up Questions

1. What specific property do you want the set $A \in P(C)$ to satisfy?
2. Are you familiar with the concept of power sets and uniqueness proofs in set theory?
3. Would you like me to walk through the concept of unique existence proofs in a general context?
4. Would you prefer an example to help illustrate this proof?
5. Is this part of a larger mathematical question or problem you're working on?

Tip: When proving uniqueness, it often helps to first prove existence (that such a set exists), and then prove that no other set could satisfy the same condition.