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No, that's not how it works.

This is how it works: (kingcounty.gov)

V = Set of Caccinated People U = Set of Unvaccinated People

V={[those ill with COVID-19 * time ill], [those severely ill with COVID-19 * time ill]} U ={[those ill with COVID-19 * time ill], [those severely ill with COVID-19 * time ill]}

Σ:V < Σ:U

Where Σ:X = total time ill (severe or otherwise) with COVID-19.

Desired outcome: minimize illness with COVID-19 to prevent viral shedding which reduces future infections, death, hospitalizations, and severity of illness.

This is a probabilities game and the vaccines reduce the probability that you run into another person who is currently shedding enough of the virus that you become infected with it. It doesn't protect you from that person who is severely ill, however. The "protection" comes into play because Σ:V < Σ:U. That's it.

Spread the word to the vaccine fanatics so they can understand how vaccines work.