ThmDex – An index of mathematical definitions, results, and conjectures.

Result R2745 on D78: Subset

The only subset of finite set with same cardinality is the ambient set itself

Formulation 0

Then \begin{equation} E \subseteq X \left( |E| = |X| \quad \implies \quad E = X \right) \end{equation}

Formulation 1

Let $X$ be a D17: Finite set such that

(i)	$\mathcal{P}(X)$ is a D80: Power set for $X$

Then \begin{equation} E \in \mathcal{P}(X) \left( |E| = |X| \quad \implies \quad E = X \right) \end{equation}

Formulation 2

Let $X$ be a D17: Finite set such that

(i)	$E \subseteq X$ is a D78: Subset
(ii)	\begin{equation} \|E\| = \|X\| \end{equation}

Then \begin{equation} E = X \end{equation}