Transitive Class/Examples/Ordinal 3

Example of Transitive Class
Let $\O$ denote the empty set.

Consider the ordinal $3$, defined as:


 * $\mathcal 3 := \set {\O, \set \O, \set {\O, \set \O} }$

$\mathcal 3$ is transitive.

Proof
Consider the $3$ elements of $\mathcal 3$ 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 $\set {\O, \set \O, \set {\O, \set \O} }$.

The elements of $\set {\O, \set \O}$, are $\O$ and $\set \O$.

We see that both are element of $\mathcal 3$.

Thus all elements of elements of $\mathcal 3$ are themselves elements of $\mathcal 3$.

Hence by definition $\mathcal 3$ is transitive.