Talk:Proper Well-Ordering determines Smallest Elements

Am I missing something here, or does the supposition that each initial segment is a set render this theorem a triviality with a very wordy and convoluted proof? --Dfeuer (talk) 20:24, 25 December 2012 (UTC)


 * Not sure. I don't follow your reasoning on this one. --Lord_Farin (talk) 22:03, 25 December 2012 (UTC)


 * I didn't give you any to follow. Basically, this theorem should be refactored into several, each simple. Something like this:
 * Each well-ordered class has a minimal element (the special case of $S=B$).
 * Each subclass of a well-ordered class has a minimal element.
 * Minimality in subclass with respect to an endorelation is equivalent to minimality under the inducted endorelation(trivial, methinks).
 * $x$ is minimal in $B$ iff $x \in B$ and $S_B \cap B = \varnothing$. --Dfeuer (talk) 23:11, 25 December 2012 (UTC)