# Membership is Left Compatible with Ordinal Addition

## Theorem

Let $x$, $y$, and $z$ be ordinals.

Let $<$ denote membership $\in$, since $\in$ is a strict well-ordering on the ordinals.

Then:

- $x < y \implies \paren {z + x} < \paren {z + y}$

## Proof

The proof proceeds by transfinite induction on $y$.

In the proof, we shall use $\in$, $\subsetneq$, and $<$ interchangeably.

We are justified in this by Transitive Set is Proper Subset of Ordinal iff Element of Ordinal.

### Base Case

\(\displaystyle \neg x\) | \(=\) | \(\displaystyle \O\) | Definition of Empty Set |

The conclusion:

- $x < \O \implies \paren {z + x} < \paren {z + \O}$

follows from propositional logic.

### Inductive Case

\(\displaystyle x < y\) | \(\leadsto\) | \(\displaystyle \paren {z + x} < \paren {z + y}\) | Hypothesis | ||||||||||

\(\displaystyle x < y^+\) | \(\leadsto\) | \(\displaystyle x < y \lor x = y\) | Definition of Successor Set | ||||||||||

\(\displaystyle \paren {z + y^+} = \paren {z + y}^+\) | \(\) | \(\displaystyle \) | Definition of Ordinal Addition | ||||||||||

\(\displaystyle \paren {z + y} < \paren {z + y^+}\) | \(\) | \(\displaystyle \) | Ordinal is Less than Successor | ||||||||||

\(\displaystyle x < y\) | \(\leadsto\) | \(\displaystyle \paren {z + x} < \paren {z + y^+}\) | Hypothetical Syllogism | ||||||||||

\(\displaystyle x = y\) | \(\leadsto\) | \(\displaystyle \paren {z + x} < \paren {z + y^+}\) | Substitutivity of Equality | ||||||||||

\(\displaystyle x < y^+\) | \(\leadsto\) | \(\displaystyle \paren {z + x} < \paren {z + y^+}\) | Proof by Cases |

### Limit Case

\(\displaystyle \) | \(\) | \(\displaystyle \forall w < y: \paren {x < w \implies \paren {z + x} < \paren {z + w} }\) | |||||||||||

\(\displaystyle \) | \(\leadsto\) | \(\displaystyle \paren {\exists w < y: x < w \implies \exists w < y \paren {z + x} < \paren {z + w} }\) | Predicate Logic Manipulation | ||||||||||

\(\displaystyle \) | \(\leadsto\) | \(\displaystyle \paren {x < y \implies \paren {z + x} < \bigcup_{w \mathop \in y} \paren {z + w} }\) | Membership of Indexed Union | ||||||||||

\(\displaystyle \) | \(\leadsto\) | \(\displaystyle \paren {x < y \implies \paren {z + x} < \paren {z + y} }\) | Definition of Ordinal Addition |

$\blacksquare$

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 8.4$