How is the atonement sufficient for salvation according to Calvinists?

Because there is not a single verse in the Bible that explicitly states that Christ laid down his life **only** for believers (the elect), Calvinists typically defend their doctrine of limited atonement by claiming the following argument: 1) The atonement is sufficient for salvation. 2) If Christ shed his blood (made atonement) for all men, then all men would be saved. 3) It is not the case that all men are saved. Therefore, Christ did not shed his blood for the sins of all men but for the elect only. In light of many clear and explicit Bible verses which state that Christ did, in fact, taste death for every man, all men, whole world, etc, Arminians typically state that the atonement was made *provisionally* for all men at the cross, but it only becomes *effective* if/when one believes. It seems like the Arminian agrees with the first premise of the limited atonement argument -- that the atonement is sufficient for salvation. In order to avoid universalism, the Arminian is forced to conclude the the blood-atonement is merely provisional rather than actual. But what if both Calvinists and Arminians are wrong on this point? Is it possible that Christ made actual atonement for all men, but not all men are saved? This leads to my question: What is the basis for arguing that the atonement is sufficient for salvation? Do Calvinists use biblical arguments, and, if so, how do they deal with all the Bible verses which clearly state that faith is necessary for salvation? If not, then what argument can a Calvinist use to support the doctrine of limited atonement?