One of the "usual" axioms of $\sf ZF$ is the axiom of regularity which states that there if $X$ is non-empty, then there is some $y\in X$ such that $y\cap X=\varnothing$.
Suppose that such a sequence would exist. What would it mean? It means there is a function $F$ whose domain is $\omega$ and $F(n+1)\in F(n)$. Using replacement, take $X$ to be the range of $F$, then there is no element of $X$ which is disjoint from it. Contradiction.
If one omits the axiom of regularity, then the answer is negative. It is consistent that there are such decreasing chains.
One should point out, however, that it is possible that there is a model of $\sf ZF$ which does have such a decreasing chain, but the function $F$ as above can never be an element of the model.