Irreflexivity of the Strict Order and Subtracting Zero in ℝ

This helper adopts the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness: $\mathbb{R}$, its order $\le$, its operations $+$ and $\cdot$, and its elements $0$ and $1$ are as in The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (ii); $<$ is the strict relation associated with $\le$, and for $x,y\in\mathbb{R}$, $x-y$ is the difference of $x$ and $y$. Let $x$ be a real number.

(i) $x<x$ does not hold.

(ii) $x-0=x$.