Proof of Order and Arithmetic of ℝ on Classes, Continued

Proof of (i). Since $\mathbb{R}$ is an ordered field by The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (ii), Rules of Arithmetic and Order in an Ordered Field (vii) gives $0<1$, and hence $0\le1$ by The Structure of ℝ as a Field and an Ordered Ring (vii). Now The Quotient 1/x, Order Rules for Sums, and Nesting of Closed Intervals in ℝ (iv), with $x=y=1$, gives

$$1\le 1+1 .$$

Proof of (ii). Suppose $x-y=0$. By The Structure of ℝ as a Field and an Ordered Ring (viii),

$$(x-y)+y=x ,$$

so $0+y=x$. Since $0+y=y$ by Arithmetic and Order Rules in the Real Numbers, Stated for ℝ (v), we get $x=y$.