Empty Set is Small

Theorem

$\O \in U$

where $U$ is the universal class.

$\ds \exists x$

$:$

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

Axiom of the Empty Set

$\ds \leadsto \ \ $

$\ds \exists x$

$:$

$\ds \forall y: \paren {y \in x \iff y \ne y}$

$\ds \leadsto \ \ $

$\ds \exists x$

$:$

$\ds x = \set {y: y \ne y}$

$\Box$

Then:

$\ds A \in U$

$\iff$

$\ds \exists x: \paren {x = A \land x \in U}$

$\ds x$

$\in$

$\ds U$

$\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

$\blacksquare$

1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 3$: Unordered Pairs
1963: Willard Van Orman Quine: Set Theory and Its Logic: $\S 7.10$
1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 5.18$