Re: Every set x equinumerous with a set y disjoint from x?

["MoeBlee"]

David C. Ullrich wrote:
> Say S is a set. Some elements of S may be ordered
> pairs; let T be the set of y such that there exists
> x such that (x,y) is in S. Now card(T) < card(P(S));
> choose y in P(S) not equal to any element of T,
> and consider S x {y}.

Then T = range(S), which is okay (S has a range even if S is not a
relation).

But how do justify card(T) < card(P(S))?

I can justify it this way, but it uses the axiom of choice:

Az card(range(z)) =< card(z) < card(Pz).

I used the axiom of choice to derive Az card(range(z)) =< card(z).

So how did you get card(T) < card(P(S)) without the axiom of choice?

MoeBlee