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
This page is beyond the scope of ZFC, and should not be used in anything other than the theory in which it resides.
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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
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 5.19$