I don't think you understood. You need a proof for all participants, because otherwise the statement you are trying to prove is meaningless.

There is no statement involving just the winner that verifies the authenticity of the auction. You cannot say "the winner won", because "the winner" is a term that only makes sense in terms of all the auction losers. There must be proofs involving all the participants, because you at the very least need to show that all the losing bids were strictly worse than the winning bid.

"prov[ing] just the winner" doesn't make any sense, or really mean anything.