Every Ordinal is a Transitive Class
From ProofWiki
Theorem
Let $A$ be an ordinal. Then $A$ is a transitive class.
That is, every ordinal is a transitive class.
Proof
By definition $A$ is an ordinal iff:
- $\forall x \in A: \left\{{y \in A : y \subset x}\right\} = x$
Thus:
- $\forall x \in A: \forall y: \left({\left({y \in A \land y \subset x}\right) \iff y \in x}\right)$
The biconditional can be reduced to an implication, and therefore:
- $\forall x \in A: \forall y \in x: \left({y \in A \land y \subset x}\right)$
Include the logical argument demonstrating the above.
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By applying the Rule of Simplification on the $\land$ statement:
- $\forall x \in A: \forall y \in x: y \in A$
By the definition of a subset:
- $\forall x \in A: x \subseteq A$
Finally, by the definition of a Transitive Class, $Tr A$.
