The problem with using TinyCore on a P133mhz is that it uses the 2.6 kernel. The old P1 hardware might not run correctly with it, but it is worth a try.
Damn Small Linux (Which used to be TinyCore founders project) is a better choice because it uses the 2.4 kernel, but it isn't supported anymore.
I have run DSL on a P100mhz with 32mb of RAM and it ran ok, but Firefox took 2 minutes to come up. Sea Monkey worked better as a browser on the older box.
Slitax is a good choice also for old hardware.