Andrej's answer that "Almost no pen-and-paper mathematics is written in ZFC" is correct. But it's perhaps also worth noting that some pen-and-paper mathematics is written in ZFC (or, at least, something closer to ZFC than to type theory), such as much of the mathematics done by set theorists (in which phrase, of course, "set" refers traditionally to a ZFC-set).
A good deal of set theory can be rephrased more structurally to make sense in type theory, but this would be a rephrasing, and the amount of work that would be required is, I think, variable. For instance, set theorists work with ordinals defined in the von Neumann way, with $\in$ as the well-founded relation. This in particular implies that ordinals are literally unique representatives of isomorphism classes of well-ordered sets. In type theory it is more natural to work "structurally" with arbitrary well-ordered sets rather than with von Neumann ordinals. This makes some things more cumbersome because of isomorphisms that get carried around (although under univalence, these isomorphisms become equalities), but probably doesn't change things a whole lot.
On the other hand, as far as I know, no one has ever come up with a convincing structural presentation of Godel's constructible universe $L$. The only structural approaches to $L$ that I know of essentially involve building a model of ZF(C) in some way and then constructing $L$ inside that model. Note that $L$ is, as far as I know, still the only way to prove the relative consistency of the Axiom of Choice, so it is of interest even to mathematicians who don't care about ZF for its own sake. (In contrast, the symmetric/permutation methods used to prove relative consistency of not-AC, and the forcing methods that can be used to prove consistency of other properties such as GCH and not-GCH, seem much more amenable to structuralization.) [Note: this paragraph was edited to remove a false claim that GCH is not forceable; see comments.]
As another example, Scott's trick is a method used in pen-and-paper mathematics that is of interest beyond set theory, but relies on the axiom of foundation from ZFC, and so is not obviously formalizable in type theory.