The proof is an SCT, a signed document from the Log which says "I promise I logged this pre-certificate". [[ In some cases it'll be the actual certificate, but for most real world TLS certificates it's a pre-certificate with the same substantive details, if you think about it you will see why the SCT baked inside your certificate cannot mention the actual document it is baked inside ]]
Now, of course at the moment this document is created, there's no way to be sure this claim is true. The logs are a distributed system.
Twenty four hours later, (the Maximum Merge Delay, this is a global policy decision) the log should, if you interrogate it, be able to show you a log entry matching the SCT, which traces to the agreed Merkle Tree head for that log. If it cannot, in principle that log has failed and must be distrusted. This happens (to one of the dozens of public logs, somewhere) a few times per year on average.
In principle, everybody who sees the log could agree that they see the same Merkle Tree head, thus the certificate really is logged. In practice the mechanisms to ensure this works, a Gossip protocol, do not exist and have never been deployed in practical use.
It's probably fine, but without ever completing the system as designed, it does have flaws.