Event Independent from Itself When Probability 0 or 1

Theorem

An event A from a probability space Ω is independent from itself if and only if either

P(A)=0orP(A)=1.
Proof
P(AA)=P(A)P(A)P(A)=P(A)2P(A)2P(A)=0P(A)(P(A)1)=0P(A)=0orP(A)=1