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This paper describes a formalisation of the lambda-calculus in a HOL-based theorem prover using nominal techniques. Central to the formalisation is an inductive set that is bijective with the alpha-equated lambda-terms. Unlike de-Bruijn indices, however, this inductive set includes names and reasoning about it is very similar to informal reasoning with “pencil and paper”. To show this we provide a structural induction principle that requires to prove the lambda-case for fresh binders only. Furthermore, we adapt work by Pitts providing a recursion combinator for the inductive set. The main technical novelty of this work is that it is compatible with the axiom of choice (unlike earlier nominal logic work by Pitts et al); thus we were able to implement all results in Isabelle/HOL and use them to formalise the standard proofs for Church-Rosser, strong-normalisation of beta-reduction, the correctness of the type-inference …