# 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

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

are logically equivalent.

## Proof

### Definition 2 implies Definition 1

\((1):\quad\) | \(\displaystyle \) | \(\) | \(\displaystyle \forall y: y = y\) | $\quad$ Equality is Reflexive | $\quad$ | ||||||||

\((2):\quad\) | \(\displaystyle \) | \(\) | \(\displaystyle \neg \exists y: y \ne y\) | $\quad$ From $(1)$ | $\quad$ | ||||||||

\((3):\quad\) | \(\displaystyle \) | \(\) | \(\displaystyle x := \left\{ {y \ \vert \ y \ne y} \right\}\) | $\quad$ | $\quad$ | ||||||||

\((4):\quad\) | \(\displaystyle \) | \(\) | \(\displaystyle \neg \exists y: y \in x\) | $\quad$ From $(2)$ and $(3)$ | $\quad$ |

$\Box$

## Sources

- 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 0.2$