Sch. C filer with Roth 401k deferrals, Roth IRA, in excess of Sch. C comp?

[Belgarath]

Interesting question here. Suppose you have a Schedule C filer, single, who is independently wealthy, so works a little but has very little earned income - only, say, $10,000 net Schedule C after deductions and SS tax reduction.

So, defers the whole $10,000 to Roth 401(k). Can he ALSO contribute to a Roth IRA? Although it is counterintuitive, I'm having trouble finding anything that prohibits it. In spite of the Roth 401(k) deferrals, he still has earned income to report on his 1040, right? And if so, I think he can technically contribute to a Roth IRA. But it doesn't feel right! What am I missing?

Edit: I suppose, as I think about it, that this is consistent with a W-2 employee - say that an employee has $10,000 of W-2 income, and defers it all to a Roth 401(k) - the W-2 is still going to show $10,000 as taxable income, on which you could presumably contribute to a Roth IRA? I still have a feeling I'm missing something, but I can't find anything proving it is wrong.