"in the world but not of the world"

In the seventeenth chapter of the gospel of John, in the Bible, Jesus prays to God about " 'the people whom you gave me out of the world' " (John 17:6; ESV for all Bible citations hereinafter): " 'I am no longer in the world, but they are in the world . . . I have given them your word, and the world has hated them because they are not of the world, just as I am not of the world' " (vv. 11, 14). What does it mean to be (as someone once condensed it) "in the world but not of the world"?

"In the world" versus "of the world": what is the distinction? My computer's Oxford American Dictionaries provides several illuminating definitions:

world [noun; from the Old English w(e)oruld: "the age of man"; in the New Testament, a translation of the Greek kosmos: "order or world"]: the earth, together with all of its countries, peoples, and natural features
"the world": all of the people, societies, and institutions on the earth
"(wo)man of the world": a person who is experienced in the ways of sophisticated society; "the world, the flesh, and the devil": all forms of temptation to sin

in [preposition]: expressing the situation of something that is or appears to be enclosed or surrounded by something else; expressing inclusion or involvement

of [preposition]: expressing the relationship between a part and a whole; indicating an association between two entities, typically one of belonging; expressing the relationship between a category and the thing being specified which belongs to such a category; indicating the material or substance constituting something
"be of": possess intrinsically; give rise to

(Admittedly, I have done some picking-and-choosing with my definitions, and I am sure a personal bias cannot be avoided, despite my best efforts to limit definitions to those I regard as most appropriate to their use in the phrases at hand: "in the world" and "of the world" as used in the context of John 17. I invite the reader to be critical of my chosen definitions and research definitions/etymologies themselves. That being said, I am going to work with the above definitions.)

"In the world" might mean [enclosed by/surrounded by/included in/involved in] the "world," or [the earth and its order in terms of: countries/people(s)/features/(sophisticated) societies/institutions/temptations].

"Of the world" might mean [a part of/associated with/belonging to/in the category of/consisting of/intrinsically possessing to give rise to] the "world," or, to say it again, [the earth's: countries/people(s)/features/(sophisticated) societies/institutions/temptations].

"In the world": If we are in the world, the world (or perhaps we could at this point say "the way of the world") is what surrounds us, what we are immersed in, what we are inevitably involved in; the way of the world is our situation.

"Of the world": If we are not of the world, or not of the way of the world, then the world's way is something of which we are not a constituent or category, something by which we are not defined or upon which we are not dependent for our identity or being. We are not the masses, and the world's way is not our distinctive, and we do not at our essence own, yield, or contribute to worldliness.

So, if we are in the world but not of it, the way of the world is indeed the situation but not the basis for who we are. That we are of the world is inevitable yet incidental somehow; it is in some sense necessary, it seems, but not ultimate, because it must be incidental to That which we truly "are of": the true and ultimate basis for who we are. Then to what do we belong? By what are we categorized? Of what whole are we a part? And what do we possess intrinsically, what is essential to our being? The answer (if we are in the world but not of it) cannot be "the world."