Gen 37:10 "And he told [it] to his father and to his brothers, and his father rebuked him and said to him, "What is this dream that you have dreamed? Will we come I, your mother, and your brothers to prostrate ourselves to you to the ground?"
Rashi: "Will we come: Isn’t your mother (Rachel) already dead? But he (Jacob) did not know that the matters referred to Bilhah, who had raised him (Joseph) as [if she were] his mother (Gen. Rabbah 84:11). Our Rabbis, however, derived from here that there is no dream without meaningless components (Talmud Berachos 55a/b). Jacob, however, intended to make his sons forget the whole matter, so that they would not envy him (Joseph). Therefore, he said,“Will we come, etc.” Just as it is impossible for your mother, so is the rest meaningless."
The difference between a mere dream (which can have a level of prophetic content (see Talmud Berachos and the dream section 55a etc.), and a full prophecy, is that a dream can contain elements of nonesense. Prophecy is a clear message from Hashem.
Zechariah 10:2 "...dreams speak falsely..."
Devarim 18: 21-22:
"Now if you say to yourself, "How will we know the word that the L-rd did not speak?" If the prophet speaks in the name of the L-rd, and the thing does not occur and does not come about, that is the thing the Lord did not speak. The prophet has spoken it wantonly; you shall not be afraid of him."
However, a prophecy must come to pass in order for it to be true.
So the way we prove prophecy is simply by seeing if it happens.
Rambam Hilchos Yesodei HaTorah: 9:1-2
Any prophet who arises and tells us that G-d has sent him does not have to [prove himself by] performing wonders like those performed by Moses, our teacher, or like the wonders of Elijah or Elisha, which altered the natural order.
Rather, the sign of [the truth of his prophecy] will be the fulfillment of his prediction of future events, as [implied by Deuteronomy 18:21]: "How shall we recognize that a prophecy was not spoken by G-d?..."
Therefore, if a person whose [progress] in the service of G-d makes him worthy of prophecy arises [and claims to be a prophet] - if he does not intend to add [to] or diminish [the Torah], but rather to serve G-d through the mitzvot of the Torah - we do not tell him: "Split the sea for us, revive the dead, or the like, and then we will believe in you." Instead, we tell him, "If you are a prophet, tell us what will happen in the future." He makes his statements, and we wait to see whether [his "prophecy"] comes to fruition or not.
Should even a minute particular of his "prophecy" not materialize, he is surely a false prophet. If his entire prophecy materializes, we should consider him a true [prophet].
We should test him many times. If all of his statements prove true, he should be considered to be a true prophet, as [I Samuel 3:20] states concerning Samuel, "And all of Israel, from Dan to Beersheba, knew that Samuel had been proven to be a prophet unto G-d."