Hi guys, have any ideas about proving trichotomy? especially the property that exactly one holds. BTW, I can't use the definition given in class that x=y <=> (x>y and x<y) since this implies two can hold at the same time.
For clarity: if you use the symbol "<" to mean "is subset of", then you can have x"<y" and "y<x". (In this case, x and y denote sets).
I'm not sure what I wrote for the 3 order axioms, but they should be for all x,y \in F, (i) exactly one of the following is true: x<y, y<x, or x=y (ii) x<y \iff 0<(x-y) (iii) 0<x, 0<y \implies 0<(x+y) and 0<(xy).