$"$ No two electrons in an atom can have the identical set of four quantum numbers." The statement is known as

The statement expresses the Pauli exclusion principle, which asserts that no two fermions (such as electrons) can occupy identical quantum states simultaneously.

An atomic orbital may contain at most two electrons, and they must possess opposite spin quantum numbers. This principle is fundamental to electron configuration and atomic structure.

In contrast, Hund's rule governs the filling of degenerate orbitals with parallel spins before pairing occurs; the Aufbau principle describes the sequential filling of orbitals from lowest to highest energy; and the Heisenberg uncertainty principle relates to the intrinsic limits in precisely measuring conjugate variables like position and momentum.

The correct identification is therefore $\boxed{A}$.