Vanishing Ideal of Larger Subset of Affine Space is Smaller

Theorem
Let $k$ be a field.

Let $n \ge 1$ be a natural number.

Let $\mathbb A^n_k$ be the standard affine space over $k$.

Let $S \subseteq T \subseteq \mathbb A^n_k$.

Then:
 * $\map I S \supseteq \map I T$

where $\map I S$ and $\map I T$ denote the vanishing ideals of $S$ and $T$, respectively.