How to express "$a \neq b \neq c \neq d$" correctly? I would like to simply say "$a, b, c, d$ are all different" by mathematical notation. 
If I want to write all equal, then I can write $a = b = c = d$. However, if I write $a \neq b \neq c \neq d$, that does not cover $a \neq c$ and $b \neq d$ etc.
Is there a neat way to write this, or shall I go with good old 
$a \neq b\ \wedge a \neq c\ \wedge a \neq d\ \wedge b \neq c \dots$?
 A: There are a number of ways to do this, but I think the most legible one is just to say in (e.g.) English, "$a,b,c,d$ are distinct". If that doesn't work, here are some options:


*

*$a \neq b \land a \neq c \land a \neq d \land b \neq c \land b \neq d \land c \neq d$

*$|\{a,b,c,d\}| = 4$.

*If you have an ordering available (such as when $a,b,c,d$ are numbers), and $a,b,c,d$ are arbitrary, you might say $a < b < c < d$.


Edit: I would point out that if you are being very formal, $a = b = c = d$ also doesn't work. If $a,b$ are elements for which an equality comparison makes sense, then $a = b$ is a proposition -- you cannot apply any further relational symbols to a proposition. The reason we "allow" $a = b = c = d$ is that everyone understands what you mean anyway.
A: Often you see this done, if the variables are subscripted, as: $x_i\ne x_j$ for $i\ne j$.  But that would require changing your notation, which, I realize, may not be desirable.
A: 
We can use the phrase $a,b,c,d$ are pairwise different to indicate that no two elements are equal.

See e.g. this  related MO post.
