Equivalence of Definitions of Empty Set

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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

\((1):\quad\) \(\displaystyle \) \(\) \(\displaystyle \forall y: y = y\) 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 $(3)$

$\Box$


Sources