Zermelo-Fraenkel, the axioms of set theory (minus choice). This would be what you were teaching if you were to start teaching mathematics by teaching its (classical) foundations.
I briefly checked the axioms, and if I recall correctly, Velleman's book does not delve into ZF in detail, but it is a beginner friendly book to give enough exposure to quantitative logic and set theory to feel comfortable working on proof-based mathematics such as LA (not survey version of LA, which should be called matrix algebra instead).
Sorry, I wasn't talking about Velleman's book on whatever, I was just challenging the sentiment that you should learn the "basics of proof based math" before linear algebra. Linear algebra is the basics of proof based math.
Edit: I should clarify that I believe that there are many approaches to learning mathematics, I'm sure that Velleman's book is fine, just that it isn't true that you should not try to learn from Axler, although perhaps someone who has a bit less of an axe to grind.
