Reflexive, Symmetric, Antisymmetric and Transitive Relations on a Set

In the setting of Class Theory NBG: the Axioms, Standing Conventions and Basic Notation, let $a$ be a set and let $R$ be a relation on $a$; for sets $u$ and $v$, $u\,R\,v$ means $(u,v)\in R$, as in Relations, Domain, Range, Inverse and Composition §relation.

¶ $R$ is reflexive if $u\,R\,u$ for every $u\in a$.

¶ $R$ is symmetric if, for all $u,v\in a$, $u\,R\,v$ implies $v\,R\,u$.

¶ $R$ is antisymmetric if, for all $u,v\in a$, $u\,R\,v$ and $v\,R\,u$ together imply $u=v$.

¶ $R$ is transitive if, for all $u,v,w\in a$, $u\,R\,v$ and $v\,R\,w$ together imply $u\,R\,w$.