Cardinal Inequality implies Ordinal Inequality

Theorem
Let $T$ be a set.

Let $\card T$ denote the cardinal number of $T$.

Let $x$ be an ordinal.

Then:


 * $x < \card T \iff \card x < \card T$

Sufficient Condition
By Cardinal Number Less than Ordinal, it follows that $\card x \le x$.

So if $x < \card T$, then $\card x < \card T$.

Necessary Condition
Suppose $\card T \le x$.

Then:

By the Rule of Transposition:
 * $\neg \card T \le \card x \implies \neg \card T \le x$

By Ordinal Membership is Trichotomy:
 * $x < \card T \iff \card x < \card T$