أَنَا سَيِّدُ وَلَدِ آدَمَ وَلَا فَخْرَ
I am the master of Adam's children, without boasting.
The report explicitly affirms the Prophet's precedence over Adam's descendants. Its expression children of Adam, however, does not literally include Adam himself. Therefore this wording alone is not a formally complete proof that the Prophet surpasses Adam.
Other evidence is brought to establish the broader doctrine. A sound argument must not claim more from an expression than its linguistic scope warrants. The legitimacy of a conclusion and the precision of one particular proof must be examined separately.