Is it enough to prove that $x \not \succsim^* y$?
Yes, it is. Step by step:
$\lnot \exists B \in A (x,y \in A \land x \in C(B))$ is equivalent to
$\forall B \in A \lnot(x,y \in B \land x \in C(B))$ is equivalent to
$\forall B \in A (x,y \notin B \vee x \notin C(B))$ is equivalent to
$\forall B \in A (x,y \in B \implies x \notin C(B))$.
Reference: How to prove it, by Velleman