Definition:Transitive Class

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

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

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

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

Notation
In order to indicate that a class $A$ is transitive, this notation is often seen:
 * $\operatorname {Tr} A$

whose meaning is:
 * $A$ 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.

Thus a class which is not transitive can be considered to be a class with "holes" in it.

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