Class Member of Class Builder

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

\(\displaystyle A \in \left\{ {x : P \left({x}\right)}\right\}\)

\(\implies\)

\(\displaystyle \exists x \in \left\{ {x : P \left({x}\right)}\right\}: A = x\)

Definition of class membership

\(\displaystyle \)

\(\implies\)

\(\displaystyle \exists x: \left({x = A \land P \left({x}\right)}\right)\)

Definition of bounded existential quantifier

\(\displaystyle \)

\(\implies\)

\(\displaystyle \exists x: \left({x = A \land P \left({A}\right)}\right)\)

Substitutivity of Class Equality

$\blacksquare$

Also see

• Set Definition by Predicate

Sources

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