# Equivalence of Definitions of Empty Set

## Theorem

The two axiomatic definitions of the empty set:

$\exists x: \forall y: \left({\neg \left({y \in x}\right)}\right)$

and

$\exists x: \forall y \in x: y \ne y$

## Proof

### Definition 2 implies Definition 1

 $(1):\quad$ $\displaystyle$  $\displaystyle \forall y: y = y$ Equality is Reflexive $(2):\quad$ $\displaystyle$  $\displaystyle \neg \exists y: y \ne y$ From $(1)$ $(3):\quad$ $\displaystyle$  $\displaystyle x := \left\{ {y \ \vert \ y \ne y} \right\}$ $(4):\quad$ $\displaystyle$  $\displaystyle \neg \exists y: y \in x$ From $(2)$ and $(3)$

$\Box$