Closure and chain conditions of Cohen forcing

The union of a descending chain of length $\gamma <\kappa$ is a function. Regularity makes its domain, a union of $\gamma$ many sets of size below $\kappa$, again have size below $\kappa$. It is therefore a common lower bound.

Let $\rho =2^{<\kappa }$ and take ${\rho }^{+}$ conditions. For $\kappa >\omega$, F2 thins their domains to a ${\rho }^{+}$-sized delta system with root $r$; for $\kappa =\omega$, use F3. There are at most $2^{\mathrm{\mid }r\mathrm{\mid }}\le \rho$ root restrictions, so two conditions agree on $r$. Their union is a function and a common extension, contradicting antichainhood. Thus the order is ${\rho }^{+}$-cc.

At $\kappa =\omega$, finite domains and the finite delta-system theorem give ccc directly. If $2^{<\kappa }=\kappa$, step 1.2 reads ${\kappa }^{+}$-cc. The thinning and pigeonhole steps use AC.