If $V \subseteq W$ and dim$(V)$=dim$(W)$, then $V=W$ If $V,W$ are subspaces of $\mathbb{R^n}$.
My proof simply states that if the dimension, $m$, of both subspaces are the same, then we know that $m$ linearly independent vectors in both $V$ and $W$ span $V$ and $W$. Thus, if $V \subseteq W$, these $m$ linearly independent vectors must be the same, and so the span of both subspaces must be the same $\Rightarrow$ $V = W$.
Is this the type of rigor expected in such a proof? I can see conceptually why the claim is true, but I'm not really sure how to organize my thoughts in such a way that "proves" the claim.

Extend a basis $v_1,\dotsc,v_m$ of $V$ to a basis $v_1,\dotsc,v_{m+n}$ of $W$. Then 
$$
m=\dim V=\dim W=m+n
$$
so $n=0$. It follows that $v_1,\dotsc,v_m$ is also a basis for $W$. Hence $V=W$.
