I have enough information to give a clear answer.

Mezlocillin is an antipseudomonal (ureidopenicillin) — a class of extended-spectrum penicillins. It is specifically susceptible to penicillinase (β-lactamase) destruction.

The penicillin classes, in terms of penicillinase resistance, are:

Mezlocillin belongs to the ureidopenicillin subgroup of antipseudomonal penicillins. Its extended gram-negative spectrum (including Pseudomonas aeruginosa, Klebsiella, and anaerobes) comes from its acyl side chain modifications — not from β-lactamase stability. It is readily hydrolyzed by penicillinases and therefore must often be combined with a β-lactamase inhibitor (like tazobactam) to protect against β-lactamase-producing organisms.

The penicillinase-resistant penicillins are specifically the anti-staphylococcal penicillins (methicillin, nafcillin, dicloxacillin), which have a bulky side chain that sterically hinders β-lactamase binding.

— Lippincott Illustrated Reviews: Pharmacology; Dermatology 5e