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I was wondering if anyone is aware of a modern book that builds a subset of elementary number theory from Peano axioms preferably in a Principia Mathematica fashion?

Or similarly an encyclopedia of mathematical proofs in some branches of mathematics only using the language of first-order logic and ZFC but not the English language.

Wiki proofs is an encyclopedia of proofs for example, but you cannot feed it to a computer program for example.

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using a web search, this one looks interesting and maybe meets your needs:

http://us.metamath.org/mpegif/mmset.html

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