Empty Set is Small

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Theorem

$\O \in U$

where $U$ is the universal class.


Proof

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

$\Box$


Then:

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

$\Box$


Hence:

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

$\blacksquare$


Sources