Equivalence of Formulations of Axiom of Pairing

Theorem
The following formulations of the axiom of pairing in the context of axiomatic set theory are equivalent:

Strong Form implies Weak Form
Let the strong form of the axiom of pairing be assumed:

By definition of the biconditional, this can be expressed as:


 * $\forall a: \forall b: \exists c: \forall z: \paren {\paren {z = a \lor z = b \implies z \in c} \land \paren {z \in c \implies z = a \lor z = b} }$

from which, by the Rule of Simplification:


 * $\forall a: \forall b: \exists c: \forall z: \paren {z = a \lor z = b \implies z \in c}$

Thus the weak form of the axiom of pairing is seen to hold.

Weak Form implies Strong Form
Let the weak form of the axiom of pairing be assumed:

By the Axiom of Specification, let us create the set $c'$ as:


 * $c' = \set {z: \paren {z \in c} \land \paren {z = a \lor z = b} }$

By the Axiom of Extensionality, it follows that:
 * $c' = \set {a, b}$

Thus we have:
 * $\forall a: \forall b: \exists c': \forall z: \paren {z = a \lor z = b \implies z \in c'}$

and:
 * $\forall a: \forall b: \exists c': \forall z: \paren {z \in c' \implies z = a \lor z = b}$

That is:
 * $\forall a: \forall b: \exists c': \forall z: \paren {z = a \lor z = b \iff z \in c'}$

Thus the strong form of the axiom of pairing is seen to hold.