The knowledge sources do not explicitly mention whether the Cohort Builder supports an "Any Event" option for "did not do" conditions. The documentation only describes the condition format as:
"Users who did <event> more/less than <some threshold>"
This wording implies a specific event must be selected, rather than a generic "Any Event" option. [Cohorts Docs]
I'd recommend checking directly in the Cohort Builder UI to see if "Any Event" appears as a selectable option in the event dropdown, or reaching out to Mixpanel Support for a definitive answer.