# Supremum and Infimum are Unique

## Theorem

### Supremum is Unique

Let $\struct {S, \preceq}$ be an ordered set.

Let $T$ be a non-empty subset of $S$.

Then $T$ has at most one supremum in $S$.

### Infimum is Unique

Let $\struct {S, \preceq}$ be an ordered set.

Let $T$ be a non-empty subset of $S$.

Then $T$ has at most one infimum in $S$.