Advanced Microeconomic Theory by Jhele and Reny states the axioms of consumer choice: completeness, transitivity, continuity, local-non satiation, strict monotonicity, convexity and strict convexity.
It is clear that the preference relation is also reflexive. I thought I could derive it from completeness, but the book states completeness is the comparison between two distinct consumption plans.
That is,
$$\forall\vec{x},\vec{y}\in X,\text{ }\vec{x}\succsim\vec{y}\vee\vec{y}\succsim\vec{x}.$$
Shouldn't it be stated as
$$\forall\vec{x},\vec{y}\in X,\text{ }\vec{x}\succsim\vec{y}\vee\vec{y}\succsim\vec{x}\vee\vec{x}=\vec{y}?$$
Or is reflexivity implied by some of the axioms? What am I missing?
We have $\forall\vec{x},\vec{y}\in X,\text{ }\vec{x}\succsim\vec{y}\vee\vec{y}\succsim\vec{x}.$ Take some arbitrary $\vec{z}$. It follows by plugging in $\vec{x} = \vec{y} = \vec{z}$ that $\vec{z} \succsim \vec{z} \lor \vec{z} \succsim \vec{z}$, which is equivalent to saying $\vec{z} \succsim \vec{z}$. So we have proved $\forall \vec{z} (\vec{z} \succsim \vec{z})$, which is the definition of reflexivity.