Element of Universe

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Theorem

Let $A$ be a class, which may be either a set or a proper class.


Then:

$\forall A: \left({A \in U \iff \exists x: x = A}\right)$

where $U$ is the universal class.


Proof

\(\displaystyle \forall A\) \(:\) \(\displaystyle \left({A \in U \iff \exists x: \left({x = A \land x \in U}\right)}\right)\) Definition of Class Membership
\(\displaystyle \forall x\) \(:\) \(\displaystyle x \in U\) Fundamental Law of Universal Class
\(\displaystyle \implies \ \ \) \(\displaystyle \forall A\) \(:\) \(\displaystyle \left({A \in U \iff \exists x: x = A}\right)\) Conjunction with Tautology

$\blacksquare$


Sources