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
Let $A$ and $B$ be sets.

Suppose that either $A$ or $B$ is empty.

Then:

By the Axiom of the Empty Set $A \times B$ is a set.

Now suppose that $A$ and $B$ are both non-empty.

Let $a \in A$ and $b \in B$.

By Singleton of Element is Subset:
 * $\set a \subseteq A$ and $\set b \subseteq B$

Therefore by Set Union Preserves Subsets:
 * $\set {a, b} \subseteq A \cup B$

$A \cup B$ is a set by Union of Small Classes is Small.

Because $\set a \subseteq A \subseteq A \cup B$, it follows that:
 * $\set a, \set {a, b} \in \powerset {A \cup B}$

where $\powerset {A \cup B}$ is the power set of $A \cup B$.

This a set by the Axiom of Powers.

Thus:
 * $\set {\set a, \set {a, b} } \subseteq \powerset {A \cup B}$

and so by definition of the power set:
 * $\set {\set a, \set {a, b} } \in \powerset {\powerset {A \cup B} }$

It has been shown that:
 * $\set {\tuple {a, b}: a \in A \land b \in B} \subseteq \powerset {\powerset {A \cup B} }$

Applying the axiom of specification and the axiom of extension, the unique set $A \times B$ is created which consists exactly of ordered pairs $\tuple {a, b}$ such that $a \in A$ and $b \in B$.

Thus it has been demonstrated that if $A$ and $B$ are non-empty, then the cartesian product $A \times B$ exists and is non-empty.

Finally, in Subset of Cartesian Product it is demonstrated that every set of ordered pairs is a subset of the cartesian product of two sets.