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Definition df-op 3106
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 3227, opprc1b 3580, opprc2 3228, and opprc3 3581). For the justifying theorem (for sets) see opth 3570. 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 <.A, B>._2 = {{{A}, (/)}, {{B}}}, justified by opthwiener 3592, which was simplified by Kazimierz Kuratowski in 1921 to our present definition. An even simpler definition <.A, B>._3 = {A, {A, B}} is justified by opthreg 6050, but it requires the Axiom of Regularity for its justification and is not commonly used. A definition that also works for proper classes is <.A, B>._4 = ((A X. {(/)}) u. (B X. {{(/)}})), justified by opthprc 4074. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 8749. Finally, an ordered pair of real numbers can be represented by a complex number as shown by crui 7694.
Assertion
Ref Expression
df-op |- <.A, B>. = {{A}, {A, B}}

Detailed syntax breakdown of Definition df-op
StepHypRef Expression
1 cA . . 3 class A
2 cB . . 3 class B
31, 2cop 3100 . 2 class <.A, B>.
41csn 3098 . . 3 class {A}
51, 2cpr 3099 . . 3 class {A, B}
64, 5cpr 3099 . 2 class {{A}, {A, B}}
73, 6wceq 1457 1 wff <.A, B>. = {{A}, {A, B}}
Colors of variables: wff set class
This definition is referenced by:  opeq1 3213  opeq2 3214  hbop 3222  opid 3224  opprc1 3227  opprc2 3228  opex 3565  elop 3566  opi1 3567  opi2 3568  opth 3570  opeqsn 3587  opeqpr 3588  uniop 3593  op1stb 3892  xpsspw 4127  xpsspwOLD 4128  relop 4152  dmsnsnsn 4398  funopg 4481  opwf 6152  rankopb 6191  rankop 6197  tskop 6666  gruop 6695
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