How do I prove that the continuum hypotheses as stated by George Cantor (There are no sets with cardinality between the cardinality of the real and the cardinality of the rational numbers) is equivalent with:
If $X$ is an uncountable subset of $\mathbb{R}$, there exists a bijection between $X$ and $\mathbb{R}$.
On
To show this the following fact is needed: two sets have the same cardinality if and only if there is a bijection between them.
Suppose that that $X$ is an uncountable set of reals. Its cardinality is therefore larger than the rationals so by CH it must have the same cardinality as the reals. So there is a bijection between $X$ and the reals.
Conversely if $X$ is a set of reals with cardinality larger than the rationals it must be uncountable, so by the second version of the hypothesis it must biject with, and thus have the same cardinality as R. So there are no subsets of R with cardinality strictly between the rationals and the R.
There is nothing to prove. The two assertions you are comparing are:
There are no sets with cardinality between those of the rational and real numbers.
Every subset of $\mathbb R$ has cardinality either that of $\mathbb Q$ or that of $\mathbb R$.
They say the same. The first one says "there is nothing between A and B", while the second says "if something is between A and B, it should be either A or B".