I have an exercise question:-
There is a poset $(P,\sqsubseteq)$ and $X$ is a subset of $P$
(1) to prove that if $X$ contains two (different) maximal elements then $X$ has no maximum.
(2) Prove that if $(\forall x,y : x,y \in X : x \sqsubseteq y \lor y \sqsubseteq x )$ and if $X$ contains a maximal element then $X$ has a maximum.
My Attempt for (1)
Suppose $X$ has a maximum $m$.
$m_i \in X \Rightarrow m_i \sqsubseteq m \qquad \{m$ is maximum of $X\} $
We know $m \in X \Rightarrow \neg (m_i \sqsubset m) \qquad \{m_i$ is maximal$\} $
So from both these we conclude $\Rightarrow m_i=m$ which is a contradiction because $m_1 \neq m_2$
So $m_1=m=m_2$
Can anybody crosscheck my attempt and guide me through the (2)