Definition:Transitive Class

From ProofWiki
Jump to navigation Jump to search


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

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

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

$x \in A \implies x \subseteq A$


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


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.

Also see

  • Results about transitive classes can be found here.

Also see