Comparing an antichain with its uniform upper shadow

Suppose \(0\le t<n/2\) and \(\mathcal A\subseteq\mathcal P(n)\) is an antichain such that \(|A|\le t\) for all \(A\in\mathcal A\). Define \[ \mathcal A_t = \left\{ B\in\binom{[n]}{t}: \text{there is some }A\in\mathcal A\text{ with }A\subseteq B \right\}. \] Prove that \[ |\mathcal A|\le |\mathcal A_t|. \]