Transitive Class/Examples/Class with Empty Class and its Singleton

Example of Transitive Class

Let $\O$ denote the empty class.

Then the class:

$\set {\O, \set \O}$

is transitive.

Proof

Consider the elements of $\set {\O, \set \O}$ in turn.

There are no elements of $\O$.

There is exactly one elements of $\set \O$, and that is $\O$.

We see that $\O$ is itself an element of $\set {\O, \set \O}$.

Thus all elements of elements of $\set {\O, \set \O}$ are themselves elements of $\set {\O, \set \O}$.

Hence by definition $\set {\O, \set \O}$ is transitive.

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 10$ Some useful facts about transitivity