Uniqueness of endpoints of a closed bounded interval in a partial order.

Question

Let $P$ be a partial order, and let $a,b,c,d$ be elements of $P$ such that $a \leq b$ and $c \leq d$. If the interval $[a,b]=[c,d]$, then does it follow that $a=c$ and $b=d$? In case it needs clarification, the notation $[x,y]$ is shorthand for $\{p \in P| x \leq p \leq y\}$

Answer 1

Of course. $b \in [a, b] = [c, d]$ implies $b \leq d$. $d \in [c, d] = [a, b]$ implies $d \leq b$. So $b = d$ by anti-symmetry. Similarly $a = c$.

Answer 2

If $[x,y]\ne\varnothing$, it is still the case that $x=\min[x,y]$ and $y=\max[x,y]$. Therefore $\varnothing\ne [a,b]=[c,d]$ implies $a=c$ and $b=d$.