@James the Rabbit had the following important objection "I accept that the Church is the New Eve--but Mary is
also a part of the Church, so she could qualify as 'New Eve' as well."
The answer is as follows (for anyone who missed it) : Mary was not a part of the Church (sharing in His death ("I am crucified with Christ") and resurrection, "in Christ" ("for the Spirit had not yet been given"--"he who is one with the Lord is one Spirit"), which wasn't even in existence at the time in which this objection wants to insist she was "New Eve" (ie, when she became pregnant, then bore and gave birth to Jesus) any more than Paul was a part of the Church when he was persecuting the Church to death.