Lemma 43.13.4. Let $X$ be a nonsingular variety and let $W,V \subset X$ be closed subvarieties with $\dim (W) = s$ and $\dim (V) = r$. Then every irreducible component $Z$ of $V \cap W$ has dimension $\geq r + s - \dim (X)$.

**Proof.**
Since $V \cap W = \Delta ^{-1}(V \times W)$ (scheme theoretically) we conclude by Lemmas 43.13.3 and 43.13.2.
$\square$

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