How about the Stone-Weierstrass theorem? If $\mathcal{A}$ is a collection of real-valued continuous functions on the compact Hausdorff space $X$ which (1) is an algebra, (2) separates points, and (3) contains the constants, then it is dense in $C(X)$. (For complex-valued functions, add (4) closed under conjugation.)
Measure theory has the monotone class and $\pi$-$\lambda$ theorems that are of a similar nature, but there we usually assign names to the hypotheses (e.g. a $\lambda$-system, which is short for three different properties).