Empty Set is Subset of All Sets/Proof 2

Proof
$S \subseteq T$ means:
 * every element of $S$ is also in $T$

or, equivalently:
 * every element that is not in $T$ is not in $S$ either.

Thus:

which means there is no element in $S$ which is not also in $T$.

There are no elements of $\O$, from the definition of the empty set.

Therefore $\O$ has no elements that are not also in any other set.

Thus, from the above, all elements of $\O$ are all (vacuously) in every other set.