The empty set $\varnothing$ is the unique set such that
\[\forall x : x\notin\varnothing.\]The existence of $\varnothing$ is justified by Axiom of Empty Set, Axiom of Existence or Axiom of Infinity. Its uniqueness is justified by Axiom of Extensionality.
\[\lvert\varnothing\rvert = 0.\]
Proof.$\varnothing$ is a bijection from $\varnothing$ to $0$.