Empty Set is Subset of All Sets

Theorem

 * $$\varnothing$$ is a subset of every set (including itself).

Proof
$$S \subseteq T$$ means "every element of $$S$$ is also in $$T$$". 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.