I need a reference/proof for "main conjecture of Iwasawa theory => refinement of HerbrandRibet ($v_p(B_{pi}) = v_p(A_i)$, where $A_i$ denotes the $i$th eigenspace of the Galois group acting on the ideal class group mod $p$)".
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The statement and proof are given in Mazur–Wiles starting at page 216. 

