No quantifier on bounded variable This is how subset is defined in set theory:
$A \subseteq B \iff \forall x \in A \implies x \in B$.
So, for how many elements this $x$ without quantifier in $x \in B$ actually stands for? All or some (at least one, maybe all)?
Isn't $\exists x \in B$ what actually is assumed?

The quantifier is over the whole implication, so $A \subseteq B$ means:
$$\forall x: (x \in A \implies x \in B)$$
