According to Pudlák (2013, p. 204), Vopěnka's principle was originally intended as a joke: Vopěnka was apparently unenthusiastic about large cardinals and introduced his principle as a bogus large cardinal property, planning to show later that it was not consistent.
Vopěnka's principle asserts that for every proper class of binary relations (each with set-sized domain), there is one elementarily embeddable into another.
This cannot be stated as a single sentence of ZFC as it involves a quantification over classes.
A cardinal κ is called a Vopěnka cardinal if it is inaccessible and Vopěnka's principle holds in the rank Vκ (allowing arbitrary S ⊂ Vκ as "classes").
Even when restricted to predicates and proper classes definable in first order set theory, the principle implies existence of Σn correct extendible cardinals for every n. If κ is an almost huge cardinal, then a strong form of Vopěnka's principle holds in Vκ: