Subset is Compatible with Ordinal Successor/Proof 2

Theorem

Let $x$ and $y$ be ordinals and let $x^+$ denote the successor set of $x$.

Let $x \in y$.

Then:

$x^+ \in y^+$

Proof

First note that by Successor Set of Ordinal is Ordinal, $x^+$ and $y^+$ are ordinals.

Let $x \in y$.

We wish to show that $x^+ \in y^+$.

By Ordinal Membership is Trichotomy:

Either $x^+ \in y^+$, $y^+ \in x^+$, or $x^+ = y^+$.

Aiming for a contradiction, suppose $y^+ = x^+$.

Then $y \in x$ or $y = x$ by the definition of successor set.

If $y = x$ then $x \in x$, contradicting the fact that Ordinal is not Element of Itself.

If $y \in x$ then since an ordinal is transitive, $y \in y$, again contradicting Ordinal is not Element of Itself.

Thus $y^+ ≠ x^+$.

Aiming for a contradiction, suppose $y^+ \in x^+$.

By definition of successor set, $y^+ \in x$ or $y^+ = x$.

If $y^+ \in x$, then since $y^+$ and $x$ are both ordinals, $y^+ \subsetneqq x$.

Then $y \in x$.

Since $y$ is transitive, $y \in y$, contradicting Ordinal is not Element of Itself.

If instead $y^+ = x$, then $x \in y \in y^+ = x$, so the same contradiction arises because $x$ is transitive.

Thus $y^+ \notin x^+$.

So the only remaining possibility, that $x^+ \in y^+$, must hold.

$\blacksquare$