Showing a set is well-ordered

Question

Let $(C,S)$ be a well-ordered set. Let $d \notin C$.

We define the set $D=C \cup \{d\}$ and the relation $S'=S\cup (C \times \{d\})$.

Show the set $(D,S')$ is well-ordered.

Any help would be much appreciated!

Answer 1

HINT: This is one of those rare occasions that drawing the new order will help understanding, even to people [like me] that have hard times with drawing the situation. So the first thing is to draw the new order and understand where $d$ fits into the story.

Suppose that $A\subseteq D$ is non-empty. What happens if $A\cap C\neq\varnothing$? What happens if the intersection is empty? In either case, find a minimal element.