# Empty Set is Small

## Theorem

$\O \in U$

where $U$ is the universal class.

## Proof

 $\displaystyle \exists x$ $:$ $\displaystyle \forall y: \paren {\neg \paren {y \in x} }$ Axiom of the Empty Set $\displaystyle \leadsto \ \$ $\displaystyle \exists x$ $:$ $\displaystyle \forall y: \paren {y \in x \iff y \ne y}$ Equality is Reflexive $\displaystyle \leadsto \ \$ $\displaystyle \exists x$ $:$ $\displaystyle x = \set {y: y \ne y}$

$\Box$

Then:

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

$\Box$

Hence:

 $\displaystyle \set {y: y \ne y}$ $\in$ $\displaystyle U$ $\displaystyle \leadsto \ \$ $\displaystyle \O$ $\in$ $\displaystyle U$ Definition of Empty Set

$\blacksquare$