Suppose is true, and that .Originally Posted bySkyWatcher

Let be the subset of for which is false.

Then by the axiom there is a smallest element of ,

and by assumption.

Then is true and is false, which is

a contradiction of the assumption that .

Hence there is no such element and so there is no such

set . And so is true for every .

(Note here we are taking if we want to

use , we would have to start with

being true rather than in our base case.)

QED.