My book gives the Dedekind's axiom or law of continuity, only for the $<$ relation:
If $S$ and $T$ are any sets of numbers (that is $S\subseteq R$ and $T\subseteq R$) satisfying the condition:
For any $x\in S$ and any $y\in T$ we have: $x < y$ then there exists a number $z$ which the following condition holds for:
If $x\in S$ and any $y\in T$, and if $x\ne z$ and $y\ne z$, then $x < z$ and $z < y$
Then it says:
To state a general condition for the relation $R$ to be continuous in the class $K$, replace in the preceding law "$ R$" by "$K$" and the word "number" by "element of the set $K$" and "$<$" by "$R$", but how do you do that? What happens to the $\ne$ ?
If $S$ and $T$ are any sets (that is $S\subseteq K$ and $T\subseteq K$) satisfying the condition:
For any $x\in S$ and any $y\in T$ we have: $xRy$, then there exists an element $z\in K$ which the following condition holds for: if $x\in S$ and any $y\in T$, and if $x\ne z$ and $y\ne z$, then $xRz$ and $zRy$