This result comes from the manner in which the orbitals are determined for particular orbitals.
First, the principal quantum number #n# is determined. This decides to which shell the orbital belongs. #n# can have any positive integer value starting with 1.
Next, the angular momentum quantum number, #l# must be specified. #l# can be any value from zero up to #n-1#.
An orbital is a #p#=orbital if it has an angular momentum quantum number, #l# equal to 1 (which implies that these orbitals first exist for quantum level #n=2#, and are found for every value of #n# after that).
Finally, for determining orbitals, the one remaining quantum number to be specified is the magnetic quantum number, #m_l#. Like each quantum number, there are restrictions on the values #m_l# can possess. In this case it is #-l, -l+1, -l+2,..., 0, 1, 2, ...l-1#.
Therefore, putting all this together: if #l=1#, (so we are referring to a #p#-orbital, the possible value for #m_l# are only #-1, 0, and +1#. These three possible values create the orbitals known as #p_x, p_y, and p_z# as the only possibilities, regardless on the value of #n#.
