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A typical argument for showing that a notion of forcing is ccc will invoke the $\Delta$-system lemma at some point, and this result has a slick proof using elementary submodels (see Lemma 24.24 of "Discovering Modern Set Theory II" by Just and Weese, or Example 2.1 of Dow's "An introduction to applications of elementary submodels to topology".

One can usually combine these two steps into a single argument to prove that a notion of forcing is ccc using elementary submodels, although I can't point to a specific published example of this at the moment. I'll see if I can find a reference for you where someone does this.

Edit: The best reference is Ramiro's answer!

2 typo

A typical argument for showing that a notion of forcing is ccc will invoke the $\delta$-system \Delta$-system lemma at some point, and this result has a slick proof using elementary submodels (see Lemma 24.24 of "Discovering Modern Set Theory II" by Just and Weese, or Example 2.1 of Dow's "An introduction to applications of elementary submodels to topology". One can usually combine these two steps into a single argument to prove that a notion of forcing is ccc using elementary submodels, although I can't point to a specific published example of this at the moment. I'll see if I can find a reference for you where someone does this. 1 A typical argument for showing that a notion of forcing is ccc will invoke the$\delta\$-system lemma at some point, and this result has a slick proof using elementary submodels (see Lemma 24.24 of "Discovering Modern Set Theory II" by Just and Weese, or Example 2.1 of Dow's "An introduction to applications of elementary submodels to topology".

One can usually combine these two steps into a single argument to prove that a notion of forcing is ccc using elementary submodels, although I can't point to a specific published example of this at the moment. I'll see if I can find a reference for you where someone does this.