# Cardinality of Power Set is Invariant

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## Theorem

Let $X, Y$ be sets.

Let $\card X = \card Y$

where $\card X$ denotes the cardinality of $X$.

Then:

- $\card {\powerset X} = \card {\powerset Y}$

where $\powerset X$ denotes the power set of $X$.

## Proof

By definition of cardinality:

- $X \sim Y$

where $\sim$ denotes the set equivalence.

Then by definition of set equivalence:

- there exists a bijection $f: X \to Y$

By definition of bijection

- $f$ is an injection and a surjection.

By Direct Image Mapping of Injection is Injection:

- the direct image mapping $\map {f^\to}: \powerset X \to \powerset Y$ is an injection.

By Direct Image Mapping of Surjection is Surjection:

- $f^\to$ is a surjection.

Then by definition of bijection:

- $f^\to: \powerset X \to \powerset Y$ is a bijection.

Hence by definition of set equivalence:

- $\powerset X \sim \powerset Y$

Thus the result by definition of cardinality:

- $\card {\powerset X} = \card {\powerset Y}$

$\blacksquare$

## Sources

- Mizar article ZFREFLE1:12