Why is the Exciting Current zero in an ideal transformer?

An ideal transformer having infinite permeability core is supposed to have zero exciting current.

An ideal transformer has infinite permeability core, which means that all the flux in the universe would want to settle inside the core, which in turn would mean that the core has an infinite flux. The relationship to exciting current is given by :-

Now given that PHI --> Infinite & mu --> Infinite, wouldn't the mathematical proof of the above equation to prove that Ie is zero be of the form :-

My question:

1. 
   
   

   Most books use the flux to be finite to prove that exciting current is zero, but fail to tell as to why they are considering the flux to be finite given that they started the concept by assuming an infinite permeability core meaning that the flux accommodated in the core would be infinite.

   
   


2. 
   

   Is there any other proof for this mathematically, or is this the right one and I have got the concept wrong. If so where exactly?