Unfortunately, the 22nd Amendment does not say that. Being elected to the office of vice-president is not being elected to the office of president.
No person shall be elected to the office of the President more than twice, and no person who has held the office of President, or acted as President, for more than two years of a term to which some other person was elected President shall be elected to the office of the President more than once.
Twelfth Amendment. You missed the point.
Amendment 12- But no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States.