Equivalence of Definitions of Ordered Pair

Theorem
The following definitions of an ordered pair are equivalent:

Equality of Ordered Pairs
From Equality of Ordered Pairs, we have that:
 * $\set {\set a, \set {a, b} } = \set {\set c, \set {c, d} } \iff a = c, b = d$

hence showing that the Kuratowski formalization fulfils the requirement for equality.

Existence of Cartesian Product
By Binary Cartesian Product in Kuratowski Formalization contained in Power Set of Power Set of Union:
 * $A \times B \subseteq \powerset {\powerset {A \cup B} }$

By Axiom of Specification, there is a set $C$ where:
 * $C = \{ x \in \powerset {\powerset {A \cup B} } : x \in A \times B \}$

Thus, the cartesian product exists.

By Cartesian Product is Unique, the cartesian product is unique.