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

Maximum element is unique

Formulation 0

Let $P = (X, {\preceq})$ be a D1103: Partially ordered set such that

(i)	$X \neq \emptyset$

Let $a, b \in X$ each be a D2218: Set element in $X$ such that

(i)	\begin{equation} \forall \, x \in X : x \preceq a \end{equation}
(ii)	\begin{equation} \forall \, x \in X : x \preceq b \end{equation}

Then \begin{equation} a = b \end{equation}

Proofs

Proof 0

Let $P = (X, {\preceq})$ be a D1103: Partially ordered set such that

(i)	$X \neq \emptyset$

Let $a, b \in X$ each be a D2218: Set element in $X$ such that

(i)	\begin{equation} \forall \, x \in X : x \preceq a \end{equation}
(ii)	\begin{equation} \forall \, x \in X : x \preceq b \end{equation}

By hypothesis, $b \preceq a$ and $a \preceq b$. A D286: Partial ordering relation is, particularly, an D289: Antisymmetric binary relation and therefore $a = b$. $\square$