Tarski-vaught and elementary substructure counter example

Question

I had to show that if $\mathcal{A} \leq \mathcal{B}, \mathcal{B} \leq \mathcal{C}, \mathcal{A} \prec \mathcal{C}, \mathcal{B} \prec \mathcal{C}$ and $\mathcal{B} \prec \mathcal{C}$, then $\mathcal{A} \prec\mathcal{B}$. It was very easy to show, however I needed to come up with a counter example that the condition $\mathcal{B} \prec \mathcal{C}$ is necessary. A hint was given as take $A = (0,1),B = [0,1]$ and $C = \mathbb{R}$ and apply the Tarski-Vaught Lemma, otherwise known as the Tarski-Vaught condition. I've taken a sequence wlog $a_1 \leq a_2 \cdots \leq a_n$ where all the $a_i$'s are in A = (0,1). By the Tarski-Vaught lemma, $\mathcal{A} \prec \mathcal{C} \iff $ for all formula $\varphi(a_1,\cdots,a_n)$, $\mathcal{C} \models \exists x \in \mathbb{R} \varphi(x,a_1,\cdots,a_n) \implies \mathcal{A} \models \exists y \in (0,1) \varphi(y,a_1,\cdot,a_n) $. But I couldn't carry on any further, any help would be appreciated.