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:

$(1)$ implies $(2)$
Let formulation $1$ of the axiom of pairing be assumed:


 * $\forall A: \forall B: \exists x: \forall y: \paren {y \in x \iff y = A \lor y = B}$

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


 * $\forall A: \forall B: \exists x: \forall y: \paren {\paren {y \in x \implies y = A \lor y = B} \land \paren {y \in x \impliedby y = A \lor y = B} }$

from which, by the Rule of Simplification:


 * $\forall A: \forall B: \exists x: \forall y: \paren {y \in x \implies y = A \lor y = B}$

Thus formulation $2$ of the axiom of pairing is seen to hold.

$(2)$ implies $(1)$
Let formulation $2$ of the axiom of pairing be assumed:


 * $\forall A: \forall B: \exists x: \forall y: \paren {y \in x \implies y = A \lor y = B}$

Thus formulation $1$ of the axiom of pairing is seen to hold.