##Haskell, 30 bytes

f s=and[x+y/=z|x<-s,y<-s,z<-s]

I think there exists a shorter solution that's more interesting, but I haven't found it.

These are 33 and 34 bytes:

f s=and$((/=)<$>s<*>)$(+)<$>s<*>s
f s|q<-((-)<$>s<*>)=all(/=0)$q$q$s

Haskell, 30 bytes

f s=and[x+y/=z|x<-s,y<-s,z<-s]

I think there exists a shorter solution that's more interesting, but I haven't found it.

These are 33 and 34 bytes:

f s=and$((/=)<$>s<*>)$(+)<$>s<*>s
f s|q<-((-)<$>s<*>)=all(/=0)$q$q$s