Proof for the bitwise XOR of two numbers to have Odd Number of set bits only when exactly one of them have odd number of bits?

Question

I was doing some Algorithms Contest Question ( OffCourse it is not live now), Where I encountered this amazing Pattern that Two Non Negative Numbers $A$ and $B$ whose Exclusive XOR represented by $C=A \oplus B$ is going to have Odd Number of Set bits Only when exactly one of them A or B have odd Number of Set bits for All other cases the C is going to have Even Number of Set Bits..
You can Take Any example of your wish A=2 and B=9
where A has odd number of set bits and B has even set Bits without even xoring them I can say $A \oplus B$ is going to have odd number of bits.
I am not able to prove it in my head that why is it true? It will appreciated if someone can provide a more concrete proof..