1. The problem statement, all variables and given/known data If a subset (call it B) of a partially ordered (by a relation R) set A has exactly one minimal element, must that element be a smallest element? Give proof or counterexample. 2. Relevant equations Well, our given, "exactly one minimal element" in PC (pred calc.) translates to: (∃b)((∀x)((x,b) ∈ R → x = b) ∧ (∀y)((∀z)((z,y) ∈ R → b = y)) i hope... 3. The attempt at a solution Call b the unique minimal element of B and let k ∈ B. Since (k,k) ∈ R, then using our assumption (∀y)((∀z)((z,y) ∈ R → b = y), b = k. Thus, (b,k) ∈ R.