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Suppose, towards a contradiction, that $G\in M$. Then $D=\mathbb{P}\setminus G$ is a ground-model set by axiom schema of separation. It is dense. If $p\notin G$, it already lies in $D$. If $p\in G$, atomless forcing order structure gives incompatible forcing conditions $q,r\le p$. Both cannot belong to the directed filter $G$, so at least one is a strengthening of $p$ in $D$.

All compatibility and extension quantifiers range over the same ground-model set $\mathbb{P}$, so this density argument is also valid internally in the transitive model $M$. Genericity requires $G\cap D\ne \varnothing$, contradicting the definition of $D$. Therefore a generic filter for an atomless order is new:The assumption is essential: a forcing atom determines a ground-model generic filter. For an atom $p$, the conditions compatible with $p$ form a generic filter $G_p\in M$: two such conditions have strengthenings below $p$, which have a common strengthening by the atom property; every dense set has a member below $p$. In a general partial order, $G_p$ need not be the principal filter above $p$.

The Rasiowa–Sikorski lemma constructs a generic filter $G_i$ over each $M_i$. Every resulting generic extension remains a countable transitive model of ZFC: there are only countably many ground-model names externally. The order $\mathbb{P}$ has the same elements and ordering at every stage, and atomless forcing order structure is absolute because all its quantifiers are bounded to $\mathbb{P}$. Hence the generic filter for an atomless order is new result gives $G_i\notin M_i$ for every $i$.

The increasing union $N={\bigcup }_{i<\omega }{M}_i$ is transitive and contains $\mathbb{P}$. If it satisfied Axiom of power set for $\mathbb{P}$, there would be a set $A\in N$ withChoose $i$ with $A\in M_i$. The next-stage generic filter $G_i$ belongs to $M_{i+1}\subseteq N$ and is an actual subset of $\mathbb{P}$. This subset assertion is absolute for the transitive set $N$, so $G_i\in A$. Transitivity of $M_i$ and $A\in M_i$ then give $G_i\in M_i$, a contradiction. Thus the power-set failure in an increasing union of generic extensions occurs already at the fixed ground-model order:The stages form an increasing chain, not an elementary chain, so the elementary chain theorem does not assert ZFC for their union.

Let $M_{n+1}=M_n[G_n]$ for a fixed atomless forcing order $\mathbb{P}$, and $N={\bigcup }_{n<\omega }{M}_n$. If a set $A\in N$ were its internal power set of $\mathbb{P}$, choose $n$ with $A\in M_n$. The next generic filter $G_n$ belongs to $N$, so $G_n\in A$; transitivity of $M_n$ implies $G_n\in M_n$, contradicting generic filter for an atomless order is new. The chain is increasing but need not be an elementary chain.