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Definition df-op 3217
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 3366, 0nelop 3735, and opprc2 3367). For the justifying theorem (for sets) see opth 3724. 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 3747, which was simplified by Kazimierz Kuratowski in 1921 to our present definition. An even simpler definition _3 is justified by opthreg 6585, 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 4235. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 9800. Finally, an ordered pair of real numbers can be represented by a complex number as shown by crui 8615.
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 3211 . 2
4 cvv 2440 . . . . 5
51, 4wcel 1528 . . . 4
62, 4wcel 1528 . . . 4
7 vx . . . . . 6
87cv 1525 . . . . 5
91csn 3208 . . . . . 6
101, 2cpr 3209 . . . . . 6
119, 10cpr 3209 . . . . 5
128, 11wcel 1528 . . . 4
135, 6, 12w3a 899 . . 3
1413, 7cab 2017 . 2
153, 14wceq 1526 1
Colors of variables: wff set class
This definition is referenced by:  dfopif 3345
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