Transitive Set is Proper Subset of Ordinal iff Element of Ordinal/Corollary

Corollary to Transitive Set is Proper Subset of Ordinal iff Element of Ordinal

Let $A$ and $B$ be ordinals.

Then:

$A \subsetneq B \iff A \in B$

Proof

We have that an ordinal is transitive.

The result follows directly from Transitive Set is Proper Subset of Ordinal iff Element of Ordinal.

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 7.8$