I'm studying Varian's Intermediate Microeconomics, in which the principle of revealed principle is stated as follows (for now we're assuming strictly convex indifference curves):
Let $(x_1, x_2)$ be the chosen bundle when prices are $(p_1, p_2)$, and let $(y_1, y_2)$ be some other bundle such that $p_1x_1+p_2x_2 \geq p_1y_1+p_2y_2$. Then if the consumer is choosing the most preferred bundle she can afford, we must have $(x_1,x_2) \succ (y_1,y_2)$.
It's further stated that "if the consumer chooses the best bundles she can afford, then the revealed preference implies preference". I can't get an intuitive grasp on this. Even if a bundle $X$ is chosen to $Y$ $(Y \neq X)$, and even if $X$ isn't the best affordable bundle, shouldn't that at least imply that $X \succ Y$? Because if not, then there must be some example where $Y \succeq X$ and still the consumer chose $X$.
Is there some counterexample in which revealed preference of $X$ does not imply preference, provided that the chosen bundle $X$ isn't the best affordable one?
