User:Dfeuer/Definition:Ordered Pair

Definition

Let $a$ and $b$ be sets.

NOTE: S&F use the angle bracket notation for ordered pairs, and use parentheses for relational structures, apparently to avoid confusion between things that are sets and things that may not be. PW seems to prefer parentheses, and as I'm lazy, I may follow PW's convention on this.

Then the ordered pair $(a, b)$ is the User:Dfeuer/Definition:Unordered Pair $\{\{a\}, \{a, b\}\}$,

where $\{a\}$ is User:Dfeuer/Definition:Singleton $a$ and $\{a, b\}$ is the unordered pair of $a$ and $b$.

By User:Dfeuer/Singleton is Set and User:Dfeuer/Axiom of Pairing, $(a,b)$ is a set.