Equality of Successors

Theorem

Let $x$ and $y$ be ordinals.

Let $x^+$ denote the successor set of $x$.

Then, $x = y \iff x^+ = y^+$

Proof

\(\ds x = y\)

\(\leadsto\)

\(\ds x^+ = y^+\)

Substitutivity of Equality

Conversely,

\(\ds x^+ = y^+\)

\(\leadsto\)

\(\ds \bigcup x^+ = \bigcup y^+\)

Substitutivity of Equality

\(\ds \)

\(\leadsto\)

\(\ds x = y\)

Union of Successor Ordinal

$\blacksquare$

Also see

• Minimally Inductive Set forms Peano Structure