They don't. The axiom of infinity gives us a chain [itex]\varnothing\in\{\varnothing\}\in\cdots\in\omega,[/itex], but [itex]\omega\in\{\omega,\cup\omega\}.[/itex] That's a chain of length [itex]\omega+1.[/itex]
Perhaps you mean [itex]a\ni b\ni c\ni\cdots[/itex], which has length less than [itex]\omega[/itex] by the axiom of regularity/foundation?