Is civil marriage in a non-religious country considered a valid, legitimate marriage, even if God's name was not mentioned in the contract and the registrar was not Muslim?

For a marriage (nikah (definition)) to be valid, four elements are required: the husband, the guardian (wali), and two just witnesses. This is based on the Prophet's (peace be upon him) saying: "There is no marriage except with a guardian and two just witnesses." The guardian must be a Muslim if the woman is Muslim, as a disbeliever has no guardianship over a Muslim. In such lands, the one responsible for Muslims acts in place of the guardian. The contract must conform to Islamic law, and there is no harm in legally documenting it to prevent problems.