Doubleton Class can be Formed from Two Sets

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Theorem

Let $V$ be a basic universe.

Let $a, b \in V$ be sets.


Then the doubleton class $\set {a, b}$ can be formed, which is a subclass of $V$.


Proof

Using the axiom of specification, let $A$ be the class defined as:

$A := \set {x: x \in V \land \paren {x = a \lor x = b} }$

That is:

$A = \set {a, b}$

By the axiom of extension, $\set {a, b}$ is the only such class which has $a$ and $b$ as elements.


Also see


Sources