What is the assumption behind "indifference curve does not cross"?
If only weak-ordering and continuity is assumed, ICs can definitely intersect.


This is not true. First, if you're speaking of indifference curves, you'd already be assuming either local non-satiation or monotonicity. 

Let's speak of indifference sets instead. The analogue of two sets, $I_1$ and $I_2$, "crossing" each other can be formalized as $I_1\ne I_2$ and $I_1\cap I_2\ne \varnothing$. 

Take two alternatives $x_1,x_2\in X$ and define two indifference sets as follows:
\begin{align}
I_1:=\{x\in X:x\sim x_1\}\quad\text{and}\quad I_2:=\{x\in X:x\sim x_2\}. 
\end{align}
WLOG, assume that $x_1\succ x_2$, so that $I_1\ne I_2$. If we allow $I_1$ to "cross" $I_2$, then $I_1\cap I_2$ must be non-empty. Let $\bar x\in I_1\cap I_2$ be an element in the intersection. By the transitivity of the indifference relation $\sim$, we have $x_1\sim \bar x$ and $\bar x\sim x_2$, implying $x_1\sim x_2$. But this is contradictory to our assumption that $x_1\succ x_2$. The contradiction therefore suggests that transitivity, a property of the weak ordering, is violated.