On [p.110](https://arxiv.org/pdf/1612.09375.pdf) Leinster says that the minimum of $x,y\in \mathbb R$ satisfies $$\min\{x,y\}\le x,\ \min\{x,y\}\le y$$ and whenever $a\in\mathbb R$ satisfies $$a\le x, a\le y,$$
 we have $a\le \min\{x,y\}$.

The last part looks counterintuitive to me. Isn't it supposed to say that whenever $a$ is less than or equal to both $x,y$, $a$ must be greater than or equal to $\min\{x,y\}$? Otherwise $a$ is "minimal", not $\min\{x,y\}$. 

Also, is Leinster's notion the same as [this definition](https://en.wikipedia.org/wiki/Maximal_and_minimal_elements#Definition) (its version for minimum) on Wikipedia?