No According to the Bible, the first woman was Lilith, not Eve. 
Adam and Lilith were both equally created from the earth.
But Because Lilith refused to submit to Adam, she abandoned (or was expelled from?) Eden. Adam was heartbroken so God created Eve from his rib to relieve his pain. Adam, Lilith, and Eve are all archetypal creatures that exist only in our psyche but influence our personality and behavior.