A property holding after all smaller ordinals holds for every ordinal. I want the notation, the mechanism, and the failure mode visible at the same time.
Definitions first
Cardinality compares sets through bijections rather than geometry. The notation \(|A|\le|B|\) means an injection \(A\hookrightarrow B\) exists, while equality requires a bijection.
The formulas should not be merged too early. The datum \(\mathsf D\), the conclusion \(\mathsf C\), and the bridge \(\Longrightarrow\) have three different logical jobs.
A small case in full
Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.
The reusable statement
What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.
A nearby false statement
Infinite cardinal arithmetic does not follow finite intuition. Removing one element or doubling a countably infinite set does not change its cardinality.
The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.