Theorem – Empty set is contained in every set.

Theorem

Let $A$ be a set. Then,

$\emptyset\subseteq{A}$

Proof

From the definition of a subset, we must show that $x\in{\emptyset}\Rightarrow{x\in{A}}$ . But from the definition of an implication, we have that the statement $x\in{\emptyset}\Rightarrow{x\in{A}}$ must be true, since the antecedent is false for all $x$ .