Does $\displaystyle X\subseteq Y$ logically imply xεX and xεY ??
In this matter I absolutely agree with emakarov's reply.
$\displaystyle \left( {X \subseteq Y} \right) \Leftrightarrow \left( {\forall x} \right)\left[ {x \in X \to x \in Y} \right]$.
Ask your protagonists does imply that $\displaystyle \left( {\forall x} \right)\left[ {x \in X \wedge x \in Y} \right]$