Empty Set is Subset of All Sets

Theorem
The empty set $\varnothing$ is a subset of every set (including itself).

That is:
 * $\forall S: \varnothing \subseteq S$

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 $\varnothing$, from the definition of the empty set.

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

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