Epsilon-induction
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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 x P[x]](http://upload.wikimedia.org/math/6/8/5/6858e811a9e4bb7f02f83ef9a69c0443.png)
This principle, sometimes called the axiom of induction (in set theory), 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 historically developed from the Greek letter ε.
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