if a ≤ b and b ≤ a then a = b (antisymmetry)
if a ≤ b and b ≤ c then a ≤ c (transitivity)
a ≤ b or b ≤ a (totalness)

if a≤b and not a=b then a≤c and not a=c and c≤b and not b=c

b≤a and not b=a and a≤c and not a=c