Hi,
If your dream did not wake you up how did you remember your dream? We can remember dreams only if it wakes us up. However we may not remember waking up! We might have a dream in the middle of the night that wakes us, go back to sleep wake up in the morning and remember the dream, however we might not remember waking up and going back to sleep.
The subconscious mind is incredibly complex. Dreams can only be understood if you watch a lot of people dream over along time. Even after studying dreams, mine and other peoples, I sometimes have problems determining whether something really happened or I dreamt it!
Perhaps in your case you were already awake when you heard your alarm clock?