Unbounded subset of ordinals a set?

[RWood]

Let R be the class of all ordinals. If a subset C of R is unbounded (i.e. for any ordinal \alpha \in R, there is \beta in C with \beta greater than \alpha ), then it seems to me that C cannot be a set, only a class. Is this true, and if so, how does one prove it? My reading on the general subject matter is limited to a bit of web browsing - perhaps the problem is trivial.

 

[RWood]

Let R be the class of all ordinals. If a subset C of R is unbounded (i.e. for any ordinal \alpha \in R, there is \beta in C with \beta greater than \alpha ), then it seems to me that C cannot be a set, only a class. Is this true, and if so, how does one prove it? My reading on the general subject matter is limited to a bit of web browsing - perhaps the problem is trivial.

I think I have the outline of a proof (there may of course be something much quicker!).

1) It is quite easy to get a 1-1 correspondence between C and R; a map C=>R is obvious; a 1-1 map R=>C can be constructed by transfinite induction, using
the unboundedness of C to ensure successor elements (or limit ordinals) are mapped to an increasing sequence of C-members.

2) On the other hand, if C is a set then it is bijective with some ordinal A (and some cardinal as well). But then A would be bijective with R, and that is clearly impossible. All this assumes we are a ZFC world.