I read all 30 previous answers, and then did "search" on this page with my browser, and
to my surprise I did not find Picard's name.
Picard's proof of the Picard Little Theorem certainly qualifies for this list.
See, for example Littlewood's Miscellany, where he discusses the question, "Can a
PhD thesis consist of one line?"
Picard's one-line proof started an enormous body of literature in XX century, beginning with
Nevanlinna theory and including Hyperbolic groups.
To be sure, Picard's original paper (CR 88(1879)1024-7) is slightly longer than one line,
but the proof itself (assuming the background that was well-known in 1879) is really
one line, as reproduced in Littlewood:-)
A slight generalization of this is called Picard's Great Theorem, the only theorem that I know, which
has the word "Great" in its standard name:-)
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