Why is Antigone so insistent on burying the body of her brother?

In Sophocles' play Antigone, the titular Antigone has lost two brothers, Eteocles and Polyneices, to the fighting in Thebes' civil war. Creon, the new ruler of Thebes, has made an edict that Eteocles will be honored with an appropriate military burial, while Polyneices, the rebellious one, will be shamed by being left on the battlefield for carrion to pick at and will be denied the sanctity of holy rites.
When Antigone summons her sister, Ismene, to her side to ask for help in secretly burying Polyneices, Ismene recoils at the danger, protesting that the "new law" forbids doing so. Antigone simply responds, "He is my brother. And he is your brother, too." Antigone believes that she is honoring her family and, in doing so, honoring the law of the gods rather than the law of man.