This is more a question of the methadology one should use to solve these type of questions:
Say there is a set $V \subseteq X \subseteq Y$ and $U \subseteq Y$ such that $$X \setminus V = U \cap X $$ Prove that $$ V = X\cap (Y \setminus U)$$
The easiest approach I found to prove these solutions is to simply draw a venn diagram that fits all the properties and then the equivalence becomes obvious but I don't think that is rigorous enough.
The other approach I know of is to do something like this:
$$x\in V \implies (x\in X) \wedge (x\notin U\cap X ) \implies (x\notin U\cap V) \implies x\notin U\implies x\in Y\setminus U\implies V\subseteq X\cap (Y\setminus U)$$
$$x\in X\cap (Y\setminus U)\implies (x\in X)\wedge(x\notin U)\implies x\notin X\setminus V \implies x\in V\implies X\cap (Y\setminus U)\subseteq V$$
However I don't like this approach because it looks so messy. Long ago I took a class on Digital Logic and we had all sorts of rules in which we could open up statements in a systematic way. De Morgan's Laws in that case did not depend on whether or not one set was a subset of another. Is there some kind of similiar methadology that one could use in this case to solve such problems in a systematic way?