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Show that a skeletal dagger preorder is just a discrete preorder, and hence just a set.
A dagger preorder is a preorder obeying the symmetry axiom:
$$ x \le y \Leftrightarrow y \le x $$ Also recall the skeletal preorders (Remark 1.26) and discrete preorders (Example 1.27):
Remark 1.26 (Partial orders are skeletal preorders).
A preorder is a partial order if we additionally have that 3. \( x \cong y \) implies \( x = y \).
Example 1.27 (Discrete preorders).
Every set \(X\) can be made into a discrete preorder. This means that the only order relations on \(X\) are of the form \( x \le x \); if \( x \neq y \) then neither \( x \le y \) or \( y \le x \) hold.