The statement is quite wrong. The stages of Sotāpanna and Cūḷasotāpanna are not rebirths, but attainments that are possible in this very life.
Arahantship is also attainable, but extremely rare. For a lay person to attain it would be ever rarer, but theoretically possible.
Being a monk, or a lay person, is just a choice of vocation, and has little to do with spiritual attainments.
Monks and nuns can be greedy, angry, conceited, deluded, just like lay people. Or they can be generous, kind, humble, and wise, just like lay people.
It is mental purity that leads to spiritual attainments like Stream-winning (sotāpanna),
or lesser Stream-winning (cūḷasotāpanna).
The Progress of Insight enumerates the requisite stages that anyone must pass through, and of those a Lesser Stream-winner would, I believe, have attained at least Purification by Overcoming Doubt
, while a Stream-winner has attained right up to Path Knowledge
and Fruition Knowledge follows on inevitably from that.