Off-diagonal long-range order: Meissner effect and flux quantization

There has been a proof by Sewell that the hypothesis of off-diagonal long-range order in the reduced density matrix ${\mathrm{\rho }}_2$ implies the Meissner effect. We present in this paper an elementary and straightforward proof that not only the Meissner effect but also the property of magnetic flux quantization follows from the hypothesis. It is explicitly shown that the two phenomena are closely related, and phase coherence is the origin for both.