The Order on Omega

In the setting of Class Theory NBG: the Axioms, Standing Conventions and Basic Notation, let $\omega$ be as in The Class Omega of Natural Numbers with Zero §omega. The two classes below are formed by class abstraction with the parameter $\omega$; their formulas quantify over set variables only, the ordered pair being a defined set symbol, so they are predicative as Class Theory NBG: the Axioms, Standing Conventions and Basic Notation §comprehension requires.

¶ The order on $\omega$ is the class

$${\le}=\{p:\exists m\,\exists n\,(p=(m,n)\wedge m\in\omega\wedge n\in\omega\wedge(m\in n\vee m=n))\},$$

and for sets $m$ and $n$, $m\le n$ means $(m,n)\in{\le}$.

¶ The strict order on $\omega$ is the class

$${<}=\{p:\exists m\,\exists n\,(p=(m,n)\wedge m\in\omega\wedge n\in\omega\wedge m\in n)\},$$

and for sets $m$ and $n$, $m<n$ means $(m,n)\in{<}$.