Corollary 2

If X is a subset of a finite set E and X $\neq$ E, then

For X is contained in the complement X$'$ of a subset of E consisting of a single element; we have $\mathrm{Card}\,(\mathrm{X}) \leqslant \mathrm{Card}\,(\mathrm{X}')$ and $\mathrm{Card}\,(\mathrm{E}) = \mathrm{Card}\,(\mathrm{X}') + 1$, hence (Proposition 2) $\mathrm{Card}\,(\mathrm{X}') < \mathrm{Card}\,(\mathrm{E})$ and a fortiori $\mathrm{Card}\,(\mathrm{X}) < \mathrm{Card}\,(\mathrm{E})$.

Definition 1 shows that, conversely, if E is a set such that

for every subset X of E such that X $\neq$ E, then E is finite.