Empty Set is Subset of All Sets

Theorem

 * $$\varnothing$$ 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.