I am currently dealing with a set theory problem where I am told to show that any well-founded partial ordering is also a well-ordering (in those words exactly, no other givens). I know I have the ability to solve this problem - the issue is, I am unsure exactly what a 'well-founded partial ordering' is. You see, by definition (at least, as I have learned them):
If $R \subseteq A \times A$ is a partial ordering, it is reflexive, transitive, and antisymmetric.
If $R \subseteq A \times A$ is well-founded, then $$\forall B \subseteq A(B \neq \emptyset \Rightarrow \exists x \in B \forall y \in B (\lnot yRx)).$$
(In other words, every nonempty subset of $A$ has a minimal element.)
The problem? For each $B$ and each minimal element $x \in B$, wouldn't this imply that $\lnot xRx$ for each of these minimal elements spanning across the various nonempty subsets of $A$? However, we stated that $R$ is a partial order, and hence reflexive (ergo $xRx$ for all $x \in A$). I see a contradiction here...
So is this entity known as a 'well-founded partial ordering' necessarily strict (read: irreflexive)? If so, why wouldn't the author have just stated that instead of calling it a 'well-founded partial ordering'?
Anyways, any clarification would be helpful from a set theory minded person. Also, not trying to be sarcastic or anything, I just don't see what good ambiguity brings to learning undergrad mathematics.