How can any one of you with a case against another dare to bring it to the unjust for judgment instead of to the holy ones? Do you not know that the holy ones will judge the world? If the world is to be judged by you, are you unqualified for the lowest law courts? Do you not know that we will judge angels? Then why not everyday matters?
If, therefore, you have courts for everyday matters, do you seat as judges people of no standing in the church? I say this to shame you. Can it be that there is not one among you wise enough to be able to settle a case between brothers? But rather brother goes to court against brother, and that before unbelievers?
Now indeed, then, it is in any case a failure on your part that you have lawsuits against one another. Why not rather put up with injustice? Why not rather let yourselves be cheated? Instead, you inflict injustice and cheat, and this to brothers.
The most charitable response would be a gentle suggestion to winter at the Saint Itarium Home for Nervous Clergy.*
I don't know that the Church in Corinth was any more receptive of this teaching than my parish church would be. For that matter, the Church Universal has seemed content in general to allow her members to use secular courts for secular matters.
But if his conclusion have not always and everywhere been followed, what are the principles St. Paul insists on?
For starters: "The holy ones will judge the world." That is, the justice of the world's laws is to be measured against the perfect justice of God's laws -- "is to be" meaning "will be, at the General Judgment," but also meaning "ought to be, by the disciple of Christ." And where the justice of the world's laws is found wanting, St. Paul teaches, for the Christian to avail himself of those laws is for the Christian to avail himself of injustice.
From this it follows, by the way, that religious matters and everyday matters are not two unrelated spheres governed by two unrelated sets of laws. The things of this world -- how we interact with our neighbors and our co-workers and our shopkeepers and our customers -- fall under God's law as much as the things of our religion, although for the most part they have different human agents who create and enforce their respective human laws.
St. Paul goes even further, though, by calling it a "failing" (NAB) or "fault" (Douay-Rheims) -- literally, which appears to mean something like "diminishing of that which should have been rendered in full measure" -- to have lawsuits between Christians in the first place. This statement wants context, surely: St. Paul himself didn't always put up with injustice from his Christian brothers.
But if you put it together with his condemnation of going to court before unbelievers, I think you get the observation that it is as hard to accept a law that is unjust toward you as it is easy to accept a law that is unjust in your favor. St. Paul would have the Corinthians understand that they ought to be so willing to do the former that they would never to the latter.
* Still, on balance I'd expect the responses to be better than if the homilist quoted without attribution the next two verses.