Basic Theorem about prime ideal

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There is a sentence in the following proof (the second proof) that says " by a basic theorem about prime ideals we have...."

What is that basic theorem? Thank you.

Intersection of prime ideals

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I guess it is the fact that the prime ideals of a ring of fractions $S^{-1}R$ correspond bijectively to the prime ideals of $R$ which do not meet $S$.