# Do the continuity axiom and transitivity axiom justify non-satiation?

Let's assume on the contrary that the indifference curve is "thick" or crosses. We can only assume the four axioms: completeness, transitivity, reflexivity and continuity. We do not assume strict monotonicity of preferences.

Define the IC as $$\{y \in X: y \sim x\}$$ where $$X$$ is a (possibly non-finite) set of consumption bundle and $$x$$ is arbitrarily chosen from $$X$$.

I do not see a way to prove this without assuming monotonicity at all.

Suppose the preference relation $$\preceq$$ is such that for all bundles $$x,y$$ we have $$x \sim y$$. This relation satisfies transitivity and continuity (it satisfies all four "axioms") but does not satisfy non-satiation.