# Element of Universe

## Theorem

Let $A$ be a class, which may be either a set or a proper class.

Then:

$\forall A: \paren {A \in U \iff \exists x: x = A}$

where $U$ is the universal class.

## Proof

 $\displaystyle \forall A$ $:$ $\displaystyle \paren {A \in U \iff \exists x: \paren {x = A \land x \in U} }$ Definition of Class Membership $\displaystyle \forall x$ $:$ $\displaystyle x \in U$ Fundamental Law of Universal Class $\displaystyle \leadsto \ \$ $\displaystyle \forall A$ $:$ $\displaystyle \paren {A \in U \iff \exists x: x = A}$ Conjunction with Tautology

$\blacksquare$