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See pages 107 and following of Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics by José Ferreirós (2008).

• Dedekind and the set-theoretical approach
  1. The algebraic origins of Dedekind's set theory, 1856-58
  2. A new fundamental notion for algebra: fields
  3. The emergence of algebraic number theory
  4. Ideals and methodology
  5. Dedekind's infinitism
  6. The diffusion of Dedekind's views

For an analysis of the proof itself, see Notes on Richard Dedekind’s "Was sind und was sollen die Zahlen?" by David Joyce (2005).