All vertebrates are chordates but all chordates are not vertebrates, why?

The members of subphylum vertebrata possess notochord during the embryonic period. So, all vertebrates are chordates. The notochord is replaced by a cartilaginous or bony vertebral column in the adult vertebrates. In protochordates, vertebral column is not formed. In urochordates, notochord is present only in tail region in larval stage and in cephalochordates, it persists throughout life. So, all chordates are not vertebrates.