Ordered Rings

In the setting of Class Theory NBG: the Axioms, Standing Conventions and Basic Notation, let $r$, with $+$, $\cdot$, $0$ and $1$, be a commutative ring, and let $\le$ be a total order on $r$.

¶ $r$, with $+$, $\cdot$, $0$, $1$ and $\le$, is an ordered ring if, for all $x,y,z\in r$: if $x\le y$ then $x+z\le y+z$; and if $0\le x$ and $0\le y$ then $0\le x\cdot y$.