Characterization of Class Membership
Jump to navigation
Jump to search
Theorem
Let $A$ and $B$ be classes.
Then:
- $\forall A, B: \paren {A \in B \iff \exists x: \paren {A = x \land x \in B} }$
where $x$ is specifically a set.
Proof
Let $V$ denote the universal class.
By Class is Subclass of Universal Class, $A \subseteq V$ and $B \subseteq V$.
By definition of universal class, every element of $V$ is a set.
Hence every element of $B$ is a set.
So if $A \in B$, then it follows that $A$ is itself a set.
Hence the result.
$\blacksquare$
Also see
- Definition:Universal Class
- Definition:Class (Zermelo-Fraenkel), where class membership is taken to be a definitional abbreviation
Sources
- 1963: Willard Van Orman Quine: Set Theory and Its Logic: $\S 6.3$