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Definition df-op 3092
Description: Kuratowski's ordered pair definition. Definition 9.1 of [Quine] p. 58. For proper classes it is not meaningful but is well-defined and we allow it for convenience (see opprc1 3225, opprc1b 3587, opprc2 3226, and opprc3 3588). For the justifying theorem (for sets) see opth 3575. 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 3599, which was simplified by Kazimierz Kuratowski in 1921 to our present definition. An even simpler definition _3 is justified by opthreg 6234, 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 4082. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 9002. Finally, an ordered pair of real numbers can be represented by a complex number as shown by crui 7911.
Assertion
Ref Expression
df-op

Detailed syntax breakdown of Definition df-op
StepHypRef Expression
1 cA . . 3
2 cB . . 3
31, 2cop 3086 . 2
41csn 3083 . . 3
51, 2cpr 3084 . . 3
64, 5cpr 3084 . 2
73, 6wceq 1414 1
Colors of variables: wff set class
This definition is referenced by:  opeq1 3208  opeq2 3209  hbop 3220  opid 3222  opprc1 3225  opprc2 3226  opex 3569  elop 3571  opi1 3572  opi2 3573  opth 3575  opeqsn 3594  opeqpr 3595  uniop 3600  op1stb 3903  xpsspw 4141  xpsspwOLD 4142  relop 4178  funopg 4523  opwf 6335  rankopb 6374  rankop 6380  tskop 6868  gruop 6901
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