Supremum and Infimum are Unique

Theorem
Let $$\left({S, \preceq}\right)$$ be a poset.

Any non-empty subset of $$S$$ admits at most one supremum and one infimum in $$S$$.

Proof
Let $$c$$ and $$c'$$ both be suprema of $$T$$ in $$S$$.

From the definition of supremum, $$c$$ and $$c'$$ are upper bounds of $$T$$ in $$S$$.

By that definition:


 * $$c$$ is an upper bound of $$T$$ in $$S$$ and $$c'$$ is a supremum of $$T$$ in $$S$$ implies that $$c' \preceq c$$;
 * $$c'$$ is an upper bound of $$T$$ in $$S$$ and $$c$$ is a supremum of $$T$$ in $$S$$ implies that $$c \preceq c'$$.

So $$c' \preceq c \land c \preceq c'$$ thus $$c = c'$$ by the antisymmety of a partial ordering.

A similar argument applies to establish the same of the infimum.