In the following paper, Truss and Felgner establish the independance of the order extension principle from the boolean prime ideal theorem by constructing a permutation model.
When proving that BPIT does not hold in this model, they write suppose I is a prime ideal supported by a subalgebra A. Then we will show all atoms a of A are in I, a contradiction. I might be being silly, but why is this a contradiction?
Felgner, U.; Truss, J. K., The independence of the prime ideal theorem from the order-extension principle, J. Symb. Log. 64, No. 1, 199-215 (1999). ZBL0933.03062.