Two finite dimensional vector spaces are isomorphic if and only if they have the same dimension (Proof review)

Question

I would like to know if this HALF proof is ok:

Theorem: Two finite dimensional vector spaces are isomorphic if and only if they have the same dimension.

Using these two corollaries:

Corollary 1: If $V$ and $W$ are finite-dimensional vector spaces such that $dim( V) > dim (W)$, then no linear map from $V$ to $W$ is injective.

Corollary 2: If $V$ and $W$ are finite-dimensional vector spaces such that $dim (V) < dim (W)$, then no linear map from V to W is surjective.

HALF Proof: Contraposition

Let us assume $V$ and $W$ are isomorphic. That means there is an invertible linear map $T$ between them. Because $T$ is invertible, in particular it is injective. So there exists a linear map injective from $V$ to $W$ which means that $dim(V) \leq dim(W)$. Then apply the same reasoning to Corollary 2. So we get $dim(V) = dim(W)$

Thanks as always!

Answer 1

You're answer seems good!

Maybe you can explicitly mention that $\dim(V)\geq\dim(W)$ after applying corollary 2, before making your final conclusion.