How to prove that this system is an invertible system or not?
To invert your system you take the derivative in $t$
$$ \dot y(t) = x(t) - \int_{-\infty}^{t} e^{-(t-\tau)} x(\tau) d \tau$$
Then $$x(t) = \dot y(t) +  \int_{-\infty}^{t} e^{-(t-\tau)} x(\tau) d \tau$$
The problem is that the integral reduces to $y(t)$ as well.
If you look at the response, the transfer function is $1/(s+1)$ the inverse response would be $s + 1$, a system with more zeros than poles. Does it mean it is not invertible in the context you are asking it.