Zero Locus of Larger Set is Smaller

Theorem
Let $k$ be a field.

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

Let $A = k \sqbrk {X_1, \ldots, X_n}$ be the ring of polynomials in $n$ variables over $k$.

Let $I, J \subseteq A$ be subsets, and $\map V I$ and $\map V J$ their zero loci.

Let $I \subseteq J$.

Then $\map V I \supseteq \map V J$.

Proof
Assume $p \in \map V J$. Then: