Bourbaki define the order in the book "theory of sets" by : Let $R(x,y)$ be a order relation if :
1) $R(x,y)$ and $R(y,z)\implies R(x,z) $
2) $R(x,y)$ and $R(y,x)\implies x=y $
3) $R(x,y)\implies R(x,x) $ and $R(y,y)$
Why we add the third condition ?
Thank you .
The idea is that an order relation is a relation that behaves like $\leq$. In particular, for any $x$, $x\leq x$ should be true (since it's $\leq$, not $<$).
Now you might ask, why does condition (3) say that if $R(x,y)$ then $R(x,x)$ and $R(y,y)$, rather than just saying $R(x,x)$? The answer is that the relation $R$ doesn't make sense for arbitrary $x$. For instance, if $R$ were the relation $\leq$ on real numbers, $R(x,x)$ isn't literally true for all $x$, only for $x\in\mathbb{R}$. So condition (3) is written to say that for any $x$ or $y$ that the relation $R$ knows how to talk about (in particular, any $x$ or $y$ such that $R(x,y)$ is true), $R(x,x)$ and $R(y,y)$ are true.
A more modern definition would instead define an order relation $R$ on a set $X$ to be a relation $R\subseteq X\times X$ satisfying conditions (1), (2), and such that $R(x,x)$ for all $x\in X$.