The following text is from Velleman's How to Prove book from the reflexive closure section:
According to the definition we gave in the last section, the relation $L = \{(x, y) \in R × R \mid x ≤ y\}$ is a total order on $R$, but the relation $M = \{(x, y) ∈ R × R | x < y\}$ is not because it is not reflexive. Of course, these relations are closely related. It’s clear that $M \subseteq L$, and the only ordered pairs in $L$ that are not in M are pairs of the form (x, x), for some x ∈ R.
My doubt is with respect to the last sentence. Isn't the only ordered pairs in $L$ which are not in M is $\forall x \in R((x,x) \in L)$. Then why is it said for some $x \in R$ rather than forall $x \in R$ ?
Somehow its both: The ordered pairs in $L\setminus M$ are all pairs for which there exists an $x$ such that the pair is $(x,x)$. The "all" is dropped in the original formulation from "(all) pairs of the form" and the "exists" is contained in "for some $x$".