Did Christ ever pray for the non-elect?

The fifth line of evidence that Reymond offers is the common observation that I have heard several times from high Calvinists. Jesus only prays for the elect. He doesn't pray for the non-elect. This is a sure sign that Jesus did not intend to die for the sins of the non-elect. Otherwise, there would exist an inconsistency between his work of intercession and his sacrificial work. Sure enough, In John 17, Jesus prays,

9 I pray for them: I pray not for the world, but for them which thou hast given me; for they are thine. 10 And all mine are thine, and thine are mine; and I am glorified in them.

My response: Isn't this partially an argument of silence? How are we to know WHY Jesus did not pray for non-believers at this point? One of the reasons offered to me was that Jesus would in no way pray for the non-elect, because those would be unanswered prayers, and it is inconceivable that any of Jesus' prayers would go unanswered by His Father.
David observes: And who were the ones given there in John 17? The 11 disciples are the ones given. The disciples were not a reference to the elect as a total class. And secondly, in verses 21-22 he does pray for the world, as follows:

20 “My prayer is not for them alone. I pray also for those who will believe in me through their message, 21 that all of them may be one, Father, just as you are in me and I am in you. May they also be in us so that the world may believe that you have sent me.

Werner: Jesus is praying that through the unity of the ELECT, the world may believe in Him! This is an incredible thing. Here is a distinction between ALL those who will believe in Jesus and the world in general. Jesus prays for the unity of all believers (i.e. the elect) so that the world may believe! Reymond's evidence is non-existent at this point.