Let $\kappa$ be an uncountable cardinal.

Given a set of S of cardinality $\kappa$, I want to construct a chain {$S_\lambda : \lambda \in \kappa$ } such that:

1) Each $S_\lambda$ is a proper subset of S with cardinality $\kappa$ and

2) For any two ordinals $\alpha < \beta$, $S_\alpha - S_\beta$ has cardinality $\kappa$

Now, using axiom of choice this is easy.

Simply take $\kappa$ many copies of S and index them using some well ordering of $\kappa$.

There is a bijection between this collection and S.

Now just remove one copy at a time and use the bijection to get the corresponding subsets of S. This gives us the chain we want.

My question is, can we always obtain such a chain in ZF without choice ?