I am trying to understand the use of quantifiers within the definition of a subset. The definition of a subset is:
$$ A \subseteq B \equiv \forall x(x \in A \rightarrow x \in B) $$
I am confused about, when you negate the statement of a subset, to an existential quantifier:
$$ \forall x(x \in A \rightarrow x \in B) \equiv \exists x(x \in A \land x \notin B) $$
How do these two have the same meaning? I am struggling to get to grips with the negated version.
To clarify, I understand the conversion and can do that applying the relevant laws but in idiomatic English it makes no sense. For Universal Quantifier, it means For all $x$, if $x$ is a member of $A$, then $x$ is a member of $B$. However, for a Existential Quantifier: There exists some value of such that $x$ is a member of $A$ but is not a member of $B$? That surely goes against the principle of a subset?