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Definition df-op 3095
Description: Kuratowski's ordered pair definition. Definition 9.1 of [Quine] p. 58. As the behavior of the usual Kuratowski definition is not very useful for proper classes, we define it to be empty in this case (see opprc1 3238, 0nelop 3600, and opprc2 3239). For the justifying theorem (for sets) see opth 3589. There are other ways to define ordered pairs; the basic requirement is that two ordered pairs are equal iff their respective members are equal. In 1914 Norbert Wiener gave the first successful definition _2 , justified by opthwiener 3611, which was simplified by Kazimierz Kuratowski in 1921 to our present definition. An even simpler definition _3 is justified by opthreg 6267, but it requires the Axiom of Regularity for its justification and is not commonly used. A definition that also works for proper classes is _4 , justified by opthprc 4097. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 9144. Finally, an ordered pair of real numbers can be represented by a complex number as shown by crui 7964.
Assertion
Ref Expression
df-op
Distinct variable groups:   ,   ,

Detailed syntax breakdown of Definition df-op
StepHypRef Expression
1 cA . . 3
2 cB . . 3
31, 2cop 3089 . 2
4 cvv 2327 . . . . 5
51, 4wcel 1416 . . . 4
62, 4wcel 1416 . . . 4
7 vx . . . . . 6
87cv 1413 . . . . 5
91csn 3086 . . . . . 6
101, 2cpr 3087 . . . . . 6
119, 10cpr 3087 . . . . 5
128, 11wcel 1416 . . . 4
135, 6, 12w3a 899 . . 3
1413, 7cab 1905 . 2
153, 14wceq 1414 1
Colors of variables: wff set class
This definition is referenced by:  dfopif 3217
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