Zero Locus of Larger Set is Smaller

Theorem
Let $k$ be a field.

Let $n\geq1$ be a natural number.

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

Let $I,J \subseteq A$ be subsets, and $V \left({I}\right)$ and $V(J)$ their zero loci.

Let $I \subseteq J$.

Then $V(I) \supseteq V(J)$.