Complex sentence: Von Neumann used the Burali-Forti paradox to prove by contradiction that the class of all ordinals is a proper class, and then he applied the axiom of limitation of size to well-order the universal class.
Split into simpler sentences: Von Neumann used the Burali-Forti paradox to prove by contradiction that the class of all ordinals is a proper class. Then he applied the axiom of limitation of size to well-order the universal class.