Complement of Strict Total Ordering
We need to show that $\not \prec$ is a weak total ordering.
First we check in turn each of the criteria for an ordering:
Let $a \not \prec b$ and $b \not \prec c$.
Now, suppose that $a \prec c$.
As $b \not \prec c$, from the Trichotomy Law, one of the following is the case:
- $b = c$ and so $a \prec b$;
- $c \prec b$ and $a \prec b$ from transitivity of $\prec$.
Both are an immediate contradiction.
So $a \not \prec c$ and so $\not \prec$ is transitive.
Suppose $a \not \prec b$.
Then from the Trichotomy Law, either $a = b$ or $b \prec a$.
Similarly, if $b \not \prec a$ we have that either $a = b$ or $a \prec b$.
As $\prec$ is asymmetric, $b \prec a \iff \neg a \prec b$.
So if $a \not \prec b$ and $b \not \prec a$, it follows that $a = b$.
So $\not \prec$ is antisymmetric.