Equivalences of Well-ordered Relation

Definitions

$ \langle A, R \rangle $ is an totally ordered class iff

1. (R is a relation on A) $ R\subseteq A\times A $
2. (irreflexivity) $ \forall x \in A\, \langle x,x \rangle \notin R $
3. (transitivity) $ \forall x,y,z \in A\, \langle x,y \rangle \in R \wedge \langle y,z \rangle \in R \rightarrow \langle x,z \rangle \in R $
4. (trichotomy) $ \forall x,y \in A\, \langle x,y \rangle \in R \vee \langle y,x \rangle \in R \vee x=y $