Harville ("Matrix algebra from a statistician's perspective", 1997, p.383, Theorem 17.2.4) states as follows:
Let $A$ represent an $m \times n$ matrix and $B$ an $m \times p$ matrix. If $\mathcal{C}(A)$ and $\mathcal{C}(B)$ are $essentially$ $disjoint$, then $ rank(A, B) = rank(A) + rank(B) $
where $\mathcal{C}(A)$ denotes the column space of $A$ and $essentially$ $disjoint$ means that $\mathcal{C}(A) \cap \mathcal{C}(B)=0$.
My question is how the condition $\mathcal{C}(A) \cap \mathcal{C}(B)=0$ can be rephrased in a way that is easy to understand. Is $\mathcal{C}(A) \cap \mathcal{C}(B)=0$ equivalent to $Ax \neq By$ for any $x \neq 0$ and $y \neq 0$?