Suppose $R$ is a partial order on set $A$, and $B \subseteq A$. Prove
If B has a smallest element, then this smallest element is unique. Thus, we can speak of the smallest element of B rather than a smallest element.
Suppose b is the smallest element of B. Then b is also a minimal element of B, and it is the only minimal element.
Suppose $x \in B$ and $x$ is smallest element. Suppose $x$ is not unique. Then there is one more smallest element, call it $y$, where $x≠y$.
Since $x$ is smallest element, $(x,y) \in R$
And since $y$ is smallest element, $(y,x) \in R$
Since $x ≠ y$, and $(x,y), (y,x)$ both in $R$, we conclude that $R$ is symmetric. But we know that $R$ is partial order, which implies it must antisymmetric. Hence we have a contradiction, which means $x$ must be unique.
Suppose $b$ is the smallest element.
Suppose $b$ is not a minimal element. It means that there is some $x$, where $x ≠ b$ and $(x,b) \in R$. But since $(b,x) \in R $, it would imply that $R$ is symmetric, but we know it's not. Hence $b$ is a minimal element.
Suppose $b$ is not the only minimal element. Then there exists one more minimal element, say $x$, such that $x ≠ b$. It follows that neither $(x,b) \in R$ nor $(b,x) \in R$. But this is a contradiction, since we know that $b$ is a smallest element, and by definition of smallest element we have
$$\forall x \in B (bRx)$$
Which implies that $(b,x) \in R$. Hence $b$ is the only minimal element. $\Box$
Is it correct?