Why these are equivalent? Situation: operator theory, spectrum of a operator.
We consider this as definition:

$\lambda$ is a eigenvalue if $\lambda x=Tx$ for some $x\ne 0$ 

but I see someone saying this:

$\lambda x-Tx=0\not \Rightarrow x=0 $ so $\lambda $ is a eigenvalue.

I cannot  see why the latest sentence implies $\lambda$ is a eigenvalue. Some help? Is this a very basic logic problem?

Note that $P\not\Rightarrow Q$ is the same as $P\land\neg Q$. But still this alternative formulation is ugly in my opinion as one must be careful with the implicit quantifiers.
