I want to show that a meet-semilattice (i.e. a poset such that any finite subset has a meet) is a semilattice as defined algebraically (binary operation with associativity, commutativity, idempotency).
For commutativity and idempotency it is obvious. For associativity I want to use the fact that $\wedge \{ a,b,c \}$ exists and doesn't depend on how we take $a,b$ and $c$.
So my question is : is my last argument correct and sufficient ? Or is it wrong ?