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

## Properties of ZF Models

Grzegorz Bancerek
Warsaw University, Bialystok

### Summary.

The article deals with the concepts of satisfiability of ZF set theory language formulae in a model (a non-empty family of sets) and the axioms of ZF theory introduced in [8]. It is shown that the transitive model satisfies the axiom of extensionality and that it satisfies the axiom of pairs if and only if it is closed to pair operation; it satisfies the axiom of unions if and only if it is closed to union operation, etc. The conditions which are satisfied by arbitrary model of ZF set theory are also shown. Besides introduced are definable and parametrically definable functions.

#### MML Identifier: ZFMODEL1

The terminology and notation used in this paper have been introduced in the following articles [10] [9] [7] [11] [12] [5] [4] [1] [13] [6] [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] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Journal of Formalized Mathematics, 1, 1989.
[5] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[6] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[7] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.
[8] Andrzej Mostowski. \em Constructible Sets with Applications. North Holland, 1969.
[9] Andrzej Trybulec. Enumerated sets. Journal of Formalized Mathematics, 1, 1989.
[10] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[11] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[12] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[13] Edmund Woronowicz. Relations defined on sets. Journal of Formalized Mathematics, 1, 1989.