Is Pluto a planet?

Lets use discrete mathematics to prove or disprove the above:

Pluto = Dog (refer to Disney)
&
Pluto = Planet

Applying Law of Transitivity (eg if A = B, & B = C, then A = C),
therefore
Dog = Planet

But a dog is not a planet! And Pluto is a dog!

Thus, by contradiction, Pluto is not a planet.