Empty Set is Small

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Theorem

$\O \in U$

where $U$ is the universal class.


Proof

\(\displaystyle \exists x\) \(:\) \(\displaystyle \forall y: \paren {\neg \paren {y \in x} }\) Axiom of the Empty Set
\(\displaystyle \leadsto \ \ \) \(\displaystyle \exists x\) \(:\) \(\displaystyle \forall y: \paren {y \in x \iff y \ne y}\) Equality is Reflexive
\(\displaystyle \leadsto \ \ \) \(\displaystyle \exists x\) \(:\) \(\displaystyle x = \set {y: y \ne y}\)

$\Box$


Then:

\(\displaystyle A \in U\) \(\iff\) \(\displaystyle \exists x: \paren {x = A \land x \in U}\) Definition of Class Membership
\(\displaystyle x\) \(\in\) \(\displaystyle U\) Fundamental Law of Universal Class
\(\displaystyle \leadsto \ \ \) \(\displaystyle A \in U\) \(\iff\) \(\displaystyle \exists x: x = A\)

$\Box$


Hence:

\(\displaystyle \set {y: y \ne y}\) \(\in\) \(\displaystyle U\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \O\) \(\in\) \(\displaystyle U\) Definition of Empty Set

$\blacksquare$


Sources