The pair of $a$ and $b$, written as
\[\{a,b\},\]is the unique set $X$ such that
\[\forall x :\enspace x\in X \enspace\lrimp\enspace x=a\lor x=b.\]The existence of $X$ is justified by Axiom of Pairing and its uniqueness is justified by Axiom of Extensionality.
For any $a$ and $b$,
\[\{a,b\} = \{b,a\}.\]
For any $a\neq b$,
\[\big\lvert\{a,b\}\big\rvert = 2.\]
Proof.$\{(a,0),(b,1)\}$ is a bijection from $\{a,b\}$ to $2$.