The Fermi paradox proves that...

...inter-galactic travel is impossible (for other sentient beings would have visited the Milky Way)
...the Milky Way is populated by only one intelligent species (for other sentient beings would have colonized the Milky Way)
...our means to detect detailed signs of intelligence outside the Milky Way are inappropriate.
...in a large enough scale, intelligent activities and natural phenomena are indistinguishable.