A "book" that satisfies all of your criteria is a set of notes from the Journal of Inquiry Based Mathematics called "Introduction to Proof" by Ron Taylor. linky
The chapters are
- Symbolic Logic
- Proof Methods
- Mathematical Induction
- Set Theory
- Functions and Relations
There are two appendices: one on mathematical writing and one on Style (By James Munkres).
It is a set of notes for an IBL class, so the assumption is that the students will be doing virtually all of the proofs themselves. I've never used this set of notes for teaching, but I've used others from the journal. I like them very much.
Their copyright notice allows free use and printing as long as attribution is given and no charge for the students other than printing costs. Similar sets of notes that I've used have cost the students about $6.
Others from the journal's website about intro to proof/foundations are http://www.jiblm.org/downloads/dlitem.aspx?id=17&category=mathnerdscollection http://www.jiblm.org/downloads/dlitem.aspx?id=16&category=mathnerdscollection http://www.jiblm.org/downloads/dlitem.aspx?id=14&category=mathnerdscollection
(These last three haven't been refereed by the journal, but they still gives links to them.)