# Epsilon-induction

In mathematics, ${\displaystyle \in }$-induction (epsilon-induction) is a variant of transfinite induction that 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:
${\displaystyle \forall x{\Big (}\forall y(y\in x\rightarrow P[y])\rightarrow P[x]{\Big )}\rightarrow \forall x\,P[x]}$
This principle, sometimes called the axiom of induction (in set theory), is equivalent to the axiom of regularity given the other ZF axioms. ${\displaystyle \in }$-induction is a special case of well-founded induction.
The name is most often pronounced "epsilon-induction", because the set membership symbol ${\displaystyle \in }$ historically developed from the Greek letter ${\displaystyle \epsilon }$.