Journal of Formalized Mathematics
Volume 1, 1989
University of Bialystok
Copyright (c) 1989 Association of Mizar Users

## The Contraction Lemma

Grzegorz Bancerek
Warsaw University, Bialystok
Supported by RPBP.III-24.C1.

### Summary.

The article includes the proof of the contraction lemma which claims that every class in which the axiom of extensionality is valid is isomorphic with a transitive class. In this article the isomorphism (wrt membership relation) of two sets is defined. It is based on [6].

#### MML Identifier: ZF_COLLA

The terminology and notation used in this paper have been introduced in the following articles [7] [8] [9] [4] [1] [5] [3] [2]

Contents (PDF format)

#### Bibliography

[1] Grzegorz Bancerek. A model of ZF set theory language. Journal of Formalized Mathematics, 1, 1989.
[2] Grzegorz Bancerek. Models and satisfiability. Journal of Formalized Mathematics, 1, 1989.
[3] Grzegorz Bancerek. The ordinal numbers. Journal of Formalized Mathematics, 1, 1989.
[4] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[5] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[6] Andrzej Mostowski. \em Constructible Sets with Applications. North Holland, 1969.
[7] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[8] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[9] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.