Lattice Definition: A (partially) ordered set $(A,\preceq)$ is defined to be a lattice when every two-element subset of $A$ has both a meet and a join (i.e., a greatest lower bound and a least upper bound).
Context: My question relates to the manner in which the definition above focuses on two-element subsets of $A$. In contrast, the relation $\preceq$ is well-founded when every (non-empty) subset of $A$ has a minimal element. I understand that one advantage of focusing on two-element subsets in the order-theoretic definition of a lattice allows an easier showing of equivalency to the algebraic definition of a lattice (e.g., by being able to easily define the meet binary operation as $x\wedge y:=\inf\{x,y\}$).
Question: Given an ordered set $(A,\preceq)$, is this ordered set a lattice under the definition given above if and only if every (non-empty) subset of $A$ has both a meet and a join?