# Class Member of Class Builder

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## Theorem

Let $A$ be a class.

Let $x$ be a set.

Let $P \left({x}\right)$ be a well-formed formula in the language of set theory.

Let $P \left({A}\right)$ denote the formula $P\left({x}\right)$ with all free instances of $x$ replaced with instances of $A$.

Let $\left\{{x: P \left({x}\right)}\right\}$ be a class specified using class builder notation.

Then:

- $A \in \left\{{x : P \left({x}\right)}\right\} \iff \left({\exists x: x = A \land P \left({A}\right)}\right)$

## Proof

\(\ds A \in \left\{ {x : P \left({x}\right)}\right\}\) | \(\implies\) | \(\ds \exists x \in \left\{ {x : P \left({x}\right)}\right\}: A = x\) | Definition of class membership | |||||||||||

\(\ds \) | \(\implies\) | \(\ds \exists x: \left({x = A \land P \left({x}\right)}\right)\) | Definition of bounded existential quantifier | |||||||||||

\(\ds \) | \(\implies\) | \(\ds \exists x: \left({x = A \land P \left({A}\right)}\right)\) | Substitutivity of Class Equality |

$\blacksquare$

## Also see

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 4.12$