Deriving that Vin is way larger than nVt from Shockley equation
The equation to determine the current in your circuit is transcendental. There is no closed form solution. To solve it you need to use an approximate iterative method like Newton-Raphson, or use a computer to do a numerical solution (which will likely also use an iterative algorithm).

But you already know the approximate solution. For a high enough voltage source value, the diode forward voltage is 0.6 or 0.7 V. And you know that 0.6 or 0.7 V is much higher than 25 mV. So you already have your intuitive solution.

Or you can change the circuit to use a current source instead of a voltage source. Then you know the current a priori. Say the current is 10 mA. Then you have 

$$ I = 10\ {\rm mA} = I_s \exp\left(\frac{V}{nV_T}-1\right)$$

If you take a typical value for \$I_s\$ as \$10^{-15}\ {\rm A}\$ or so (and neglect the -1 term inside the exponential), then you have

$$ \exp\left(\frac{V}{nV_T}\right) = 10^{13}$$

Then you can immediately get 

$$ \frac{V}{nV_T} \approx 30 $$

which should satisfy any requirement for \$V \gg nV_T\$.