Does CCA-security implies CPA-security While reading Katz &amp; Lindell's textbook (2nd edition) if found three main security definitions: ING-EAV-Security, ING-CPA-Security and ING-CCA-Security. (From this forum I know there are more, but these are the most basic ones). I also read, that ING-CPA-Security implies ING-EAV-Security and that ING-EAV-Security is weaker than ING-CPA-Security which is weaker than ING-CCA-Security.
My question: Does ING-CCA-Security also implie ING-CPA-Security?
My Sketch on the reduction proof:

Basic Assumption: I got a Cryptosystem #1 that is CCA secure

Based on this Cryptosystem I can build a CPA secure Cryptosystem #2, and I assume, that an adversary can break it with non negligible probability

Therefore I can build a Reduction: R can simulate the CPA game with A and just gives the messages of A to his challenger and the response back to A. R then outputs the result of A.

Because A has a non negligible probability in breaking #2, R also has a non negligible probability in breaking #1, which gives a contradiction to the basic assumption

Therefore CCA implies CPA


I hope the basic idea is correct.