Definition 1

Tag 03PD, permanent. Cite it as [Bourbaki, Tag 03PD].

Theory of Sets, chapter II, § 2. Ordered pairs

Given two sets X and Y, the set

\mathscr{E}_z((\exists x)(\exists y)(z = (x, y) \text{ and } x \in X \text{ and } y \in Y))

is called the product of X and Y and is denoted by X $\times$ Y.

The relation $z \in X \times Y$ is thus equivalent to " $z$ is an ordered pair and $\mathrm{pr}_1 z \in X$ and $\mathrm{pr}_2 z \in Y$ ". The sets X and Y are called the first and second factors of X $\times$ Y.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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