It has been claimed that if $P = NP$, then every problem in $P$ (equivalently $NP$) is NP-complete.
As a blanket statement, I find this wrong because, well, there's the trivial problem or constant time problems, but more interestingly, I expect that there are classes that are very small that are provably not NP-complete but aren't entirely trivial. This is in analogy to the fact that it is known that $P != EXP$, there must be a $C \subsetneq NP$, presumably $C= NTIME(\log(n))$. But I have never heard discussed such a small class.
The smallest well-known class I can think of is $AC^0$, but I don't know whether it is known that it is not $NPC$. And there's Ryan's recent result that $NEXP \not\subset ACC^0$, which leads me to believe that it would have to be a much smaller class than $AC^0$ that would be known to be disctinct from $NP$.
I think the first statement might be considered to be true if all one has are deterministic polytime reductions (which would imply that -all- problems in $P$ are $P$-complete), but that seems to be too simplistic. A particular proof might use a particular reduction, but the distinctness of complexity classes is not parameterized by those reductions (or is it?).
So, thoughts/suggestions? Should I just accept the first statement since $NPC$ is defined with respect to deterministic polytime reductions? And even so, is there a well-known non-trivial class that is provably a proper subset of $NP$?