more specific..not because of having a PPL or not, but obtain ATC clearance and aircraft must be equipped with a two-way radio for communications with ATC and an operating Mode C transponder, furthermore aircraft overflying the upper limit of any Class B airspace must have an operating Mode C transponder.
Visual Flight Rules (VFR) flights may proceed under their own navigation after obtaining clearance but must obey any explicit instructions given by ATC.
Pilots operating in Class B airspace must have a private pilot's certificate, or have met the requirement of CFR 61.95. These are often interpreted to mean "have an instructor's endorsement for having been properly trained in that specific Class B space." However, it does not apply to student pilots seeking sport or recreational certificates. Some Class B airports (within Class B airspaces) prohibit student pilots from taking off and landing there and are listed in the AIM .
Well, I'm pretty sure any self-respecting pilot is aware there are minimum rules and laws to satisfy with operations in any class of airspace.
My point was that you said he need an IR... which is not true. PPL is fine as long as all the requirements that you posted above are met.