We've been asked to teach ourselves a unit on ordinals for our final exam tomorrow, I grasp how to prove that certain ordinals are distinct but I am having trouble figuring out a proof to show ordinal addition is associative. All the proofs I have found online use methods that we have not covered yet. Would someone be able to guide me through a proof of associativity for ordinal addition?
Here's a general overview of what I know:
Two ordinals are equal if they are order isomorphic.
Ordinal addition for two well ordered sets $a=ord(A, <_A) b=ord(B, <_B)$ then$ a+b=ord(AUB, <_+) $ where $x <_+ y$ if either x, y are in A and $x<_A y$, or x, y are in B and $x<_B y$, or x is in A and y is in B
Thank you so much