# Definition:Transitive Class

## Definition

Let $S$ denote a class, which can be either a set or a proper class.

Then $S$ is transitive if and only if every element of $S$ is also a subset of $S$.

That is, $S$ is transitive if and only if:

$x \in S \implies x \subseteq S$

## Notation

In order to indicate that a class $S$ is transitive, this notation is often seen:

$\operatorname{Tr} S$

whose meaning is:

$S$ is (a) transitive (class or set).

Thus $\operatorname{Tr}$ can be used as a propositional function whose domain is the class of all classes.