# Equivalence of Definitions of Empty Set

## Theorem

The two axiomatic definitions of the empty set:

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

and

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

## 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 $(3)$