In our previous works, a Metatheorem in ordered fixed point theory showed that certain maximal element principles can be reformulated to various types of fixed point theorems for progressive maps and conversely. Therefore, there should be dual principles related to minimality, anti-progressive maps, and others. In the present article, we derive several minimal element principles particular to the Metatheorem and their applications. One such application is the Brøndsted–Jachymski Principle. We show that known examples due to Zorn (1935), Kasahara (1976), Brézis–Browder (1976), Tasković (1989), Zhong (1997), Khamsi (2009), Cobzaș (2011), and others can be improved and strengthened by our new minimal element principles.