I have good knowledge of Propositional and Predicate Logic. I want to begin set theory. I am looking for books that explain set theory principles using Logic. And also books which have lots of theorems with proofs on basic set theory?
A first course in logic by Shawn Hedman covers both logic and set theory. I personally like Kunen's development of elementary set theory, but that might be a bit terse. It certainly is based on logic. Jech's book contains a lot of material, but might be a bit inaccessible at first.
If you want something basic, slow paced, with detailed proofs and solutions of some exercises, then this could be a good choice.
Schaum's outline of theory and problems of set theory and related topics By Seymour Lipschutz
This book was also recommended in an answer to this related question: Set theory practice problems?
I should add that I only have this book in hand only a few times, mostly when I was looking for some exercises. But what I wrote above more-or-less characterizes the books from Schaum's Outlines series I've seen so far.
The following book, which I got in 1996 or 1997, might be suitable:
Set Theory, Logic and their Limitations by Moshe Machover (1996)
Axiomatic Set Theory by Patrick Suppes is a classic, logic-based introduction to formal set theory and it's in Dover, so it's very cheap. You'll also find a very good chapter on axiomatic set theory in Elliot Mendelson's Introduction To Mathematical Logic.
There's an interesting pedagogical debate in mathematics regarding set theory: Should it be taught with or without some grounding in mathematical logic? I learned it without logic from Enderton's classic and very readable Elements of Set Theory-but Enderton's treatment is confusing in regards to the more subtle aspects of it,such as expressing the precise difference in ZFC between elements and sets.Personally,I find formal set theory to be rather confusing without the basics of logic. Many of the formal axioms of set theory were devised specifically to overcome logical paradoxes that develop without them-so expressing them as predicates in logical arguments are probably the clearest way to express the more subtle aspects of it. Expressing some of these more difficult points in logical notation goes a long way towards clearing them up.
Levy's book on Basic Set Theory" is also very good. For an easy going and clear introduction you may go for Goderi's Classic; and somewhat more formal Vaught's Introduction or Hanjal's set theory. Kunen's 1980 work and Jech 2006 monograph are for advanced courses.