Attempting to apply more flexible, informal reasoning to predicate logic as demonstrated helpfully to me by another user in answer to my last question.
$\{x \in B \mid x \notin C\} \in \mathscr P(A)$
I read this as "For every item 'x' that's in the set B, if it's not in the set C, then it's a subset of A."
Seems like this can pretty simply be rewritten as
$\forall x \left( \left(x \in B \land x \notin C \right) \to x \in A \right)$
Is this accurate?
As noted in comments, the word "subset" is incorrect in your translation, but your formalized version is correct.
The careful way of writing this out is first to note:
$$\left\{x\in B\mid x\notin C\right\}\in \mathcal P(A)$$
is equivalent to:
$$\left\{x\in B\mid x\notin C\right\}\subseteq A$$
which is equivalent to:
$$\forall x\in \left\{x\in B\mid x\notin C\right\}:x\in A$$
which is equivalent to:
$$\forall x:\left((x\in B\land x\notin C)\implies x\in A\right)$$