# $X$ is defined as the collection of sets $X = \{X_i: i \in [0, 1] \}$ where each set has cardinality $c$

$$X$$ is defined as the collection of sets $$X = \{X_i: i \in [0, 1] \}$$ where each set has cardinality $$c$$ of the continuum and each pair is disjoint.

How do I prove that for each $$i \in [0, 1]$$, the set $$[0, 1] \times \{i\}$$ has the same cardinality as $$X_i$$(which can therefore be replaced by any cardinality)?

I know that the general idea is that the union clearly has a subset of cardinality $$c$$ and so the union must have a cardinality greater than or equal to $$c$$. So if I am able to show that this union also has cardinality less than or equal to $$c$$, I can probably use the Schröder–Bernstein Theorem to deduce the result.

The cardinality of X$$_i$$ is c.
Clearly #([0,1]×{i}) = c×1 = c = #X$$_i$$.