Ordinal Membership is Transitive
Jump to navigation
Jump to search
Theorem
Let $\On$ denote the class of all ordinals.
Then:
- $\forall \alpha, \beta, \gamma \in \On: \paren {\alpha \in \beta} \land \paren {\beta \in \gamma} \implies \alpha \in \gamma$
Proof
By Strict Ordering of Ordinals is Equivalent to Membership Relation the statement to be proved is equivalent to:
- $\forall \alpha, \beta, \gamma \in \On: \paren {\alpha \subsetneqq \beta} \land \paren {\beta \subsetneqq \gamma} \implies \alpha \subsetneqq \gamma$
which follows (indirectly) from Subset Relation is Transitive.
$\blacksquare$
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $5$: Ordinal Numbers: $\S 1$ Ordinal numbers: Corollary $1.16$