Empty Set Subset of All

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Theorem

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

That is:

$\forall S: \varnothing \subseteq S$

$S \subseteq T$ means every element of $S$ is also in $T$, or, equivalently, every element that is not in $T$ is not in $S$ either.

Thus:

	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle S \subseteq T$	$\iff$	$\displaystyle \forall x \in S: x \in T$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of a subset
	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\iff$	$\displaystyle \neg \left({\exists x \in S: \neg \left({x \in T}\right)}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	De Morgan's Laws (Predicate Logic)

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.

$\blacksquare$

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle S \subseteq T\)	\(\iff\)	\(\displaystyle \forall x \in S: x \in T\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of a subset
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\iff\)	\(\displaystyle \neg \left({\exists x \in S: \neg \left({x \in T}\right)}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	De Morgan's Laws (Predicate Logic)