1.12.3 Disequality

Finally, let us also say something about disequality, which is negation of equality:¹¹We use “inequality” to refer to $<$ and $\leq$ . Also, note that this is negation of the propositional identity type. Of course, it makes no sense to negate judgmental equality $\equiv$ , because judgments are not subject to logical operations.

(x\neq_{A}y)\ :\!\!\equiv\ \lnot(x=_{A}y).

If $x\neq y$ , we say that $x$ and $y$ are unequal or not equal. Just like negation, disequality plays a less important role here than it does in classical mathematics. For example, we cannot prove that two things are equal by proving that they are 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 $x\neq 0$ .

Title	1.12.3 Disequality
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