All of welfare economics is based on some theorems, some of which I can remember, but I am looking for a comprehensive list.
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I presume you meant The Fundamental Theorems:
The first fundamental theorem of welfare economics states that any Walrasian equilibrium is Pareto-efficient.
The second fundamental theorem of welfare economics states that, under the assumptions that every production set $Y_j$ is convex and every preference relation $\geq _i$ is convex and locally nonsatiated, any desired Pareto-efficient allocation can be supported as a price quasi-equilibrium with transfers. Further assumptions are needed to prove this statement for price equilibriums with transfers.
Or you could read Stiglitz's Paper yourself if you were so inclined.