1.12.3 Disequality

Finally, let us also say something about disequality,which is negation of equality:

$$(x{\ne }_{A}y):\equiv \mathrm{\neg }(x{=}_{A}y).$$

If $x\ne y$, we say that $x$ and $y$ are unequalor not equal.Just like negation, disequality plays a less important role here than it does in classicalmathematics. For example, we cannot prove that two things are equal by proving that theyare not unequal: that would be an application of the classical law of double negation, see §3.4 (http://planetmath.org/34classicalvsintuitionisticlogic).

Sometimes it is useful to phrase disequality in a positive way. For example,in Theorem 11.2.4 (http://planetmath.org/1122dedekindrealsarecauchycomplete#Thmprethm1) we shall prove that a real number $x$ has an inverse if,and only if, its distance from $0$ is positive, which is a stronger requirement than $x\ne 0$.