Epsilon-induction

In mathematics, ∈-induction (epsilon-induction) is a variant of transfinite induction, which can be used in set theory to prove that all sets satisfy a given property P(x). If the truth of the property for x follows from its truth for all elements of x, for every set x, then the property is true of all sets. In symbols:

$\forall x(\forall y(y\in x\rightarrow P(y))\rightarrow P(x))\rightarrow \forall xP(x)$

This principle is equivalent to the axiom of regularity. It can be converted into a transfinite induction on the rank of the set x.

The name is most often pronounced "epsilon-induction", because the set membership symbol $\in$ historically developed from the Greek letter $\varepsilon$.