there is U,W,V subspaces of $\displaystyle R^{4}$ and dimU=dimV=dimW=3

prove that $\displaystyle U\cap V\cap W\neq{0}$

how i tried to solve it:

dim(U+V)=dimU+dimV-$\displaystyle dim(U\cap V)$

because U+V is a subspace of $\displaystyle R^{4}$then $\displaystyle dim(U+V)\leq4$

so $\displaystyle dim(U\cap V)\geq10$

$\displaystyle dim(W+(U\cap V))=dimW+dim(U\cap V)-dim(W\cap U\capV)$

because $\displaystyle W+(U\cap V)$ is a subspace of $\displaystyle R^{4}$then

$\displaystyle dim(W+(U\cap V))\leq4$

by inputing the previos data i get

$\displaystyle dim(W\cap U\cap V)\geq17$

is it correct?