Correct. The good news that Elements still works otherwise, you just need to add the missing axiom.

But many other "proofs" have been found to be false. The book "Metamath: A Computer Language for Mathematical Proofs" (by Norm Megill and yours truly) is available at: https://us.metamath.org/downloads/metamath.pdf - see section 1.2.2, "Trusting the Mathematician". We list just a few of the many examples of proofs that weren't.

Sure, there can be bugs in programs, but there are ways to counter such bugs that give FAR more confidence than can be afforded to humans. Lean's approach is to have a small kernel, and then review the kernel. Metamath is even more serious; the Metamath approach is to have an extremely small language, and then re-implement it many times (so that a bug is unlikely to be reproduced in all implementations). The most popular Metamath database is "set.mm", which uses classical logical logic and ZFC. Every change is verified by 5 different verifiers in 5 different programming languages originally developed by 5 different people:

* metamath.exe aka Cmetamath (the original C verifier by Norman Megill)

* checkmm (a C++ verifier by Eric Schmidt)

* smetamath-rs (smm3) (a Rust verifier by Stefan O'Rear)

* mmj2 (a Java verifier by Mel L. O'Cat and Mario Carneiro)

* mmverify.py (a Python verifier by Raph Levien)

For more on these verifiers, see: https://github.com/metamath/set.mm/blob/develop/verifiers.md