Atomless forcing order

An order is atomless if it has no forcing atom, equivalently every condition has a pair of incompatible forcing conditions below it. For a fixed order belonging to a transitive model, this property is absolute because its quantifiers range over the unchanged set of conditions. Infinite-domain Fn forcing with at least two possible values is atomless: prescribe different values at one fresh coordinate.

If a generic filter $G$ for an atomless forcing order belonged to its ground model, then the complement $\mathbb{P}\setminus G$ would be a dense ground-model set. A condition outside $G$ is already there; a condition in $G$ has two incompatible strengthenings, at least one outside its directed filter. Genericity would then require meeting the complement, a contradiction.