Empty Set is Subset of All Sets

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

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

$$S \subseteq T \iff \forall x \in S : x \in T$$ Definition of a subset

$$\iff \lnot \left({\exists x \in S: \lnot \left({x \in T}\right)}\right)$$ ForAll-NotExistsNot

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.

Q.E.D.