partial order with chain condition does not collapse cardinals

If $P$ is a partial order which satisfies the $\kappa$ chain condition and $G$ is a generic subset of $P$ then for any , $\lambda$ is also a cardinal in $𝔐[G]$, and if $\mathrm{cf}(\alpha )=\lambda$ in $𝔐$ then also $\mathrm{cf}(\alpha )=\lambda$ in $𝔐[G]$.

This theorem is the simplest way to control a notion of forcing, since it means that a notion of forcing does not have an effect above a certain point. Given that any $P$ satisfies the ${|P|}^{+}$ chain condition, this means that most forcings leaves all of $𝔐$ above a certain point alone. (Although it is possible to get around this limit by forcing with a proper class.)