I was seeing the proof of Minimal Element in a Totally Ordered Set is unique and smallest from $\mathsf{Pr}\infty\mathsf{fWiki}$; this is how the proof begins:
Let $(S,\preceq)$ be a totally ordered set.
Let $m$ be a minimal element of $(S,\preceq)\,.$
[...]
Proof:
By definition of minimal element: $$\forall y\in S: y\preceq m\implies m= y$$ ....
This really bothered me; for the definition of minimal element, also given in $\mathsf{Pr}\infty\mathsf{fWiki}$ goes as:
Let $(S,\preceq)$ be an ordered set.
Let $T\subseteq S$ be a subset of $S\,.$
Definition 1:
An element $x\in T$ is a minimal element of $T$ iff: $$\forall y \in T: y\preceq x\implies y= x\,.$$ ....
Notice that the minimal element is defined for the subset of $S$ and not for $S\,.$
So, how did they write in the proof in the former excerpt above that $\forall y\in S: y\preceq m\implies m= y\,?$ The minimal element $m$ must be defined for a subset of $S$ and not for the whole set $S\,.$
Why did then, they do so in the proof above? Why didn't they take a subset and took all the $y$ belonging to the whole set $S\,?$
It seems they are not in the agreement with the very definition of minimal element they gave.
Could anyone shed some light on this?