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RE: Countable infinity is the smallest cardinal infinity even without assuming the Axiom of Choice (WARNING: technicalities ahead)
First of all: Thank you very much for this proof. I'll link to it form my post.
I just have one question. It's probably just an unknown notation on my side but, first you say "it implies a bijection f" and then you speak of functions 
. What does it mean in this context?
Since A and A hat have the same cardinality there exists a bijective f from A to A hat.
It is defined as follows
Ah I see. Thank you. :)