# Equivalence of Definitions of Empty Set

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## Contents

## Theorem

The two axiomatic definitions of the empty set:

- $\exists x: \forall y: \paren {\neg \paren {y \in x} }$

and

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

are logically equivalent.

## Proof

### Definition 2 implies Definition 1

\(\text {(1)}: \quad\) | \(\displaystyle \) | \(\) | \(\displaystyle \forall y: y = y\) | Equality is Reflexive | |||||||||

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

\(\text {(3)}: \quad\) | \(\displaystyle \) | \(\) | \(\displaystyle x := \set { {y: y \ne y} }\) | ||||||||||

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

$\Box$

### Definition 1 implies Definition 2

\(\text {(1)}: \quad\) | \(\displaystyle \) | \(\) | \(\displaystyle \forall y: \paren {\neg \paren {y \in x} }\) | ||||||||||

\(\text {(2)}: \quad\) | \(\displaystyle \) | \(\) | \(\displaystyle \forall y: \neg \paren {y \in x} \land \paren {y \in x} \implies y \ne y\) | Rule of Explosion | |||||||||

\(\text {(3)}: \quad\) | \(\displaystyle \) | \(\) | \(\displaystyle \forall y: \paren {\neg \paren {y \in x} \implies \paren {y \in x \implies y \ne y} }\) | Rule of Exportation, from $(2)$ | |||||||||

\(\text {(4)}: \quad\) | \(\displaystyle \) | \(\) | \(\displaystyle \forall y: \paren {y \in x \implies y \ne y}\) | Modus Ponendo Ponens, from $(1)$ and $(3)$ |

## Sources

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