Nonempty Class has Members

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Theorem

Let $A$ be a class.


Then:

$A \ne \O \iff \exists x: x \in A$


Proof

NotZFC.jpg

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\(\ds A \ne \O\) \(\leadstoandfrom\) \(\ds \neg \forall x: \paren {x \in A \iff x \in \O}\) Definition of Class Equality
\(\ds \) \(\leadstoandfrom\) \(\ds \neg \forall x: \neg x \in A\) Definition of Empty Set
\(\ds \) \(\leadstoandfrom\) \(\ds \exists x: x \in A\) De Morgan's Laws (Predicate Logic)

$\blacksquare$


Sources