Expressing the negation of a statement using predicates and quantifiers with conditions. Let $A,B$ and $C$ be the subsets of a universal set $U$. Let $A(x)$ denote the predicate "$x\in A$". 
Express the negation of 
$$A\subseteq (B\cup C)$$
using quantifiers, the predicates $A(x),B(x)$ and $C(x)$, and logical connectives such that no negation is on the immediate left of  


*

*a quantifier  

*a statement joined by a logical connective


Attempt at solution:
I ended up with $\exists x\in U(x\in A \land (x\notin B \land x\notin C))$ and am very tempted to just write it as $\exists x\in U(A(x) \land B(x) \land C(x))$ where $B(x)$ denotes "$x\notin B$" and $C(x)$ denotes "$x\notin C$".
Is there any way to express it using $B(x): x\in B$ and $C(x): x\in C$?

The conditions imposed in the statement of the problem allow you to place a negation immediately in front of a simple predicate like $B(x)$, so you can write
$$\exists x\in U\big(A(x)\land\neg B(x)\land\neg C(x)\big)\;.$$
