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
- 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: $\S 5.18$