Element of Universe

Theorem

Let $A$ be a class, which may be either a set or a proper class.

Then:

$\forall A: \paren {A \in U \iff \exists x: x = A}$

where $U$ is the universal class.

That is, if $A$ is an element of $U$, then $A$ is a set.

Proof

\(\ds \forall A: \, \)

\(\ds \leftparen {A \in U}\)

\(\iff\)

\(\ds \rightparen {\exists x: \paren {x = A \land x \in U} }\)

Characterization of Class Membership

\(\ds \forall x: \, \)

\(\ds x \in U\)

\(\)

\(\ds \)

Fundamental Law of Universal Class

\(\ds \leadsto \ \ \)

\(\ds \forall A: \, \)

\(\ds \leftparen {A \in U}\)

\(\iff\)

\(\ds \rightparen {\exists x: x = A}\)

Conjunction with Tautology

$\blacksquare$

Sources

• 1963: Willard Van Orman Quine: Set Theory and Its Logic: $\S 6.9$