TITLE: Let $(A; ≼)$ be any poset. Prove that the relation $≺ := ≼ ∩ ≠$ on A is transitive.
QUESTION [0 upvotes]: How can i prove this ?
Let $(A; ≼)$ be any poset. Prove that the relation $≺ := ≼ ∩ ≠$ on A is transitive.

REPLY [0 votes]: Let $a<b$ and $b<c$. 
Then $a\leq b$ and $b\leq c$ so that the transitivity of $\leq$ guarantees that $a\leq c$.
It remains to prove that $a\neq c$.
If $a=c$ then we have $a\leq b$ and $b\leq a$, hence $a=b$ on base of antisymmetry. 
This however contradicts that $a<b$.
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