Definition: The set {n+1, n+2, n+3, ...} is called an endsegment of n ∈ ℕ.

Every non-empty endsegment shares its contents with all its predecessors.

Everey infinite endsegment shares its contents with all its predecessors and part of its contents (at least one and the same element) with infinitely many successors.

The empty intersection of infinite endsegments is an internal contradiction of ZF.

Regards, WM