Uniqueness of Least and Greatest Elements, Properties of the Strict Order, and Trichotomy for Total Orders

In the setting of Class Theory NBG: the Axioms, Standing Conventions and Basic Notation, let $a$ be a set, $\le$ a partial order on $a$ and $<$ its associated strict relation.

¶ For every subclass $s$ of $a$, there is at most one $m\in s$ such that $m\le v$ for every $v\in s$, and at most one $m\in s$ such that $v\le m$ for every $v\in s$.

¶ For all sets $u$ and $v$, $u<v$ holds if and only if $u\le v$ and $u\neq v$.

¶ For every $u\in a$, $u<u$ does not hold.

¶ For all $u,v,w\in a$, if $u<v$ and $v<w$ then $u<w$.

¶ For all $u,v\in a$, $u\le v$ holds if and only if $u<v$ or $u=v$.

¶ If $\le$ is total, then for all $u,v\in a$ exactly one of $u<v$, $u=v$ and $v<u$ holds.

¶ If $\le$ is total, then for all $u,v\in a$, $u\le v$ fails if and only if $v<u$.