Definition:Transitive Class

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

Then $S$ is transitive every element of $S$ is also a subclass of $S$.

That is, $S$ is transitive :
 * $x \in S \implies x \subseteq S$

or:
 * $\forall x: \forall y: \paren {x \in y \land y \in S \implies x \in S}$

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 known as
A transitive class is also known as a complete class.

Also see

 * Definition:Swelled Class
 * Definition:Supercomplete Class


 * Class has Subclass which is not Element: while there is nothing stopping a class to be such that all its elements are subclasses, it is not possible for all its subclasses to be among its elements.


 * Class is Transitive iff Union is Subset