$X$ is isometrically isomorphic with $Y$, then we denote by $X \simeq Y$ and, if $X \simeq Z$ for $Z$ a subspace of $Y$, then we denote by $X \preceq Y$ (invented notation).
Prove that $X \simeq Y$, if and only if $X \preceq Y$ and $Y \preceq X$.
It actually looks very natural, but I don't know why I can't write it.
Just to make things official (and to move the discussion here in case I'm wrong):
I don't think this holds. I believe you need https://ncatlab.org/nlab/show/Cantor-Schroeder-Bernstein+theorem for this to work, but this is not true for Banach spaces.