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Definition df-op 3277
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 3426, 0nelop 3796, and opprc2 3427). For the justifying theorem (for sets) see opth 3785. 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 3808, which was simplified by Kazimierz Kuratowski in 1921 to our present definition. An even simpler definition _3 is justified by opthreg 6732, 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 4312. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 9963. Finally, an ordered pair of real numbers can be represented by a complex number as shown by crui 8772.
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 3271 . 2
4 cvv 2500 . . . . 5
51, 4wcel 1533 . . . 4
62, 4wcel 1533 . . . 4
7 vx . . . . . 6
87cv 1530 . . . . 5
91csn 3268 . . . . . 6
101, 2cpr 3269 . . . . . 6
119, 10cpr 3269 . . . . 5
128, 11wcel 1533 . . . 4
135, 6, 12w3a 899 . . 3
1413, 7cab 2077 . 2
153, 14wceq 1531 1
Colors of variables: wff set class
This definition is referenced by:  dfopif  3405
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