Definition:Transitive Class

Definition

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

Then $S$ is transitive iff every element of $S$ is also a subset of $S$.

That is, $S$ is transitive iff:

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

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.

Also see

• Class is Transitive iff Union is Subset

Sources

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 12$: The Peano Axioms