Diphthongs are actually somewhat complicated in English.
There are phonemic diphthongs /ay aw oy/ (to use Kenyon and Knott's American phonemic system), as in buy bough boy, which are definitely phonemic, and in contrast with the vowels they transition between (/a o i u/).
Then there are phonetic diphthongs, which represent diphthongized allophones of single tense vowel phonemes, like [əʊ̯] in RP or [ow] in American English for the tense mid back rounded vowel phoneme /o/.
In American English, all tense vowel phonemes have diphthongized allophones -- front tense vowels /i e/ have a high front /i/ glide, and back tense vowels /u o/ have a high back rounded /u/ glide. There are no pure tense vowel phonemes for these diphthongs to contrast with, so they are normally not represented phonemically, viz:
/i/ [iy] /u/ [uw]
/e/ [ey] /o/ [ow]
This is such a common phenomemon that it's a feature of an American accent in other languages; speakers of American English are often unable to hear or produce the distinction between (for example) /e:/ and /ei/, or /o:/ and /ou/, when speaking European languages.
Edit: I find that I have already answered this question here.