Not only does it have proofs, they're the subject of the book! You are exactly the target audience, someone who knows math but wants to make the leap to proofs.

Not only does it have proofs, they're the subject of the book! You are exactly the target audience, someone who knows math but wants to make the leap to proofs. There is essentially no calculation, it starts with a lovely introduction to logical quantifiers and sets and moves on from there. By the end you'll have constructed the Reals and experimented with everything from vector spaces to the hyperreals to the basics of galois theory. LADR is definitely a good pick for after you're comfortable with proofs. They really aren't especially hard, but I remember how tricky it was until I actually learned how to start and where to go.