Ordered pair

This representation of an ordered pair by sets satisfies $⟨a,b⟩=⟨c,d⟩$ exactly when $a=c$ and $b=d$. Its intersection recovers $\{a\}$, and its union recovers $\{a,b\}$; if the latter is a singleton then $b=a$. Its rank of a set is $\max (\mathrm{rank}(a),\mathrm{rank}(b))+2$, which bounds the ranks of function graphs in absoluteness of cardinalhood in limit ranks.