How did prussia cease to exist?
it stopped being a nation after the Austro-Prussian War of 1866. It became the dominant part of the North German federation. After the Franco-Prussian war of 1871, Prussia became the dominant part of the new German Empire. It did that because Prussia coupled with the other German states would be more important than Prussia alone. Whether Bismarck actually intended Prussia to be absorbed is an extensive historical debate.