We can easily see Antigone's stubborness in her conversation with her sister, Ismene, when she rejects Ismene's practical and pragmatic advice and chooses to listen to her own convictions and beliefs and let those be her moral compass at this particular moment in her life. Note how she responds to Ismene's suggestion that they should obey the person who is placed in power over them:
I will bury him myself.
And even if I die in the act, that death will be a glory.
I will lie with the one I love and loved by him--
an outrage sacred to the gods!
Antigone therefore reveals her stubborn determination to disregard Creon's laws and see her brother buried in response to her belief in the gods and their moral dictates. She completely rejects Ismene's practical and pragmatic approach to ensure survival and she indicates that she will follow through with her moral instincts that tell her to ignore Creon, no matter what.
Antigone's stubbornness is revealed almost every time she speaks in the play. She sets her face like flint. She knows she has transgressed Creon's public decree. She admits the deed: "I avow it; I make no denial." And she admits knowing it violated the king's order: "I knew it: could I help it? It was public." So she stubbornly stands by her action.
The Chorus points out how stubborn she is:
"The maid shows herself passionate child of passionate sire and knows not how to bend before troubles."
Creon agrees, but explains that it is possible to tame stubborn spirits, recalling high-tempered horses that have been broken. Ironically, after that statement, he exhibits his own stubbornness by declaring that he would not be a man--Antigone would--if he were to pardon her.
Antigone asks whether he has any greater punishment than death to dole out to her, and he says that will satisfy him. Antigone then states:
"Why then dost thou delay? In thy discourse there is nought that pleases me--never may there be!--and so my words must needs be unpleasing to thee."
She means that Creon's words will never convince her to change her mind, just as her words will never convince Creon to change his. This reveals not only Antigone's stubbornness, but the matching stubbornness of Creon. Because of Creon's superior rank, he may be able to execute Antigone, but he can never tame or break her--unlike the spirited horses he has seen.
"A fool convicts me of folly"