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Definition df-op 3080
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 3202, opprc1b 3557, opprc2 3203, and opprc3 3558). For the justifying theorem (for sets) see opth 3545. 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 3569, 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 6033, 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 4053. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 8743. Finally, an ordered pair of real numbers can be represented by a complex number as shown by crui 7681.
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 3074 . 2 class <.A, B>.
41csn 3071 . . 3 class {A}
51, 2cpr 3072 . . 3 class {A, B}
64, 5cpr 3072 . 2 class {{A}, {A, B}}
73, 6wceq 1428 1 wff <.A, B>. = {{A}, {A, B}}
Colors of variables: wff set class
This definition is referenced by:  opeq1 3188  opeq2 3189  hbop 3197  opid 3199  opprc1 3202  opprc2 3203  opex 3540  elop 3541  opi1 3542  opi2 3543  opth 3545  opeqsn 3564  opeqpr 3565  uniop 3570  op1stb 3871  xpsspw 4106  xpsspwOLD 4107  relop 4131  funopg 4460  opwf 6135  rankopb 6174  rankop 6180  tskop 6639  gruop 6673
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