Where can I find a rigorous proof of the Second Fundamental Theorem of Asset Pricing?
That is:
A arbitrage-free market is complete if and only if it has a unique risk neutral measure.
Please do not refer to Shreve's Stochastic Calculus For Finance II which contains a sketch of this result. Also please do not refer to the original paper.
A proof of the statement in almost full generality can be found in Chapter 2.6 of Jarrow's Continuous-Time Asset Pricing Theory (or Chapter 2.4 if you are using the first edition of this book). The rest of Chapter 2 is dedicated to the first fundamental theorem of asset pricing.
For a proof of the special case where trading at discrete times is allowed, you may see Corollary 2.2.12 in The Mathematics of Arbitrage by Delbaen and Schachermayer. The rest of the book probably constitutes the most complete treatment of the first fundamental theorem of asset pricing.
For completeness, the original paper is: Harrison, J. Michael, and Stanley R. Pliska. "A stochastic calculus model of continuous trading: complete markets." Stochastic processes and their applications 15.3 (1983): 313-316. The paper is short (only 4 pages long), but it does use a lot of machinery that is left for the reader to check in the references.