Empty Set Exists

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Theorem

$\varnothing \in U$

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \exists x$	$:$	$\displaystyle \forall y: \left({\neg \left({y \in x}\right)}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Axiom of Existence
$\displaystyle $	$\displaystyle \implies$	$\displaystyle $	$\displaystyle \exists x$	$:$	$\displaystyle \forall y: \left({y \in x \iff y \ne y}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Axiom of Subsets Equivalents
$\displaystyle $	$\displaystyle \implies$	$\displaystyle $	$\displaystyle \exists x$	$:$	$\displaystyle x = \left\{ {y: y \ne y}\right\}$	$\displaystyle $	$\displaystyle $	$\displaystyle $

$\Box$

Then:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle A \in U$	$\iff$	$\displaystyle \exists x: \left({x = A \land x \in U}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of class membership
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle x$	$\in$	$\displaystyle U$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Fundamental Law of Universal Class
$\displaystyle $	$\displaystyle \implies$	$\displaystyle $	$\displaystyle A \in U$	$\iff$	$\displaystyle \exists x: x = A$	$\displaystyle $	$\displaystyle $	$\displaystyle $

$\Box$

Hence:

	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \left\{ {y: y \ne y}\right\}$	$\in$	$\displaystyle U$	$\displaystyle $	$\displaystyle $	$\displaystyle $
	$\displaystyle $	$\displaystyle \implies$	$\displaystyle $	$\displaystyle \varnothing$	$\in$	$\displaystyle U$	$\displaystyle $	$\displaystyle $	$\displaystyle $	by definition of the empty set

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \exists x\)	\(:\)	\(\displaystyle \forall y: \left({\neg \left({y \in x}\right)}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Axiom of Existence
\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle \exists x\)	\(:\)	\(\displaystyle \forall y: \left({y \in x \iff y \ne y}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Axiom of Subsets Equivalents
\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle \exists x\)	\(:\)	\(\displaystyle x = \left\{ {y: y \ne y}\right\}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle A \in U\)	\(\iff\)	\(\displaystyle \exists x: \left({x = A \land x \in U}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of class membership
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle x\)	\(\in\)	\(\displaystyle U\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Fundamental Law of Universal Class
\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle A \in U\)	\(\iff\)	\(\displaystyle \exists x: x = A\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \left\{ {y: y \ne y}\right\}\)	\(\in\)	\(\displaystyle U\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle \varnothing\)	\(\in\)	\(\displaystyle U\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by definition of the empty set