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Definition df-op 3241
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 3357, opprc1b 3714, opprc2 3358, and opprc3 3715). For the justifying theorem (for sets) see opth 3704. 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 3726, 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 6115, 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 4203. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 8833. Finally, an ordered pair of real numbers can be represented by a complex number as shown by crui 7933.
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 3235 . 2 class <.A, B>.
41csn 3233 . . 3 class {A}
51, 2cpr 3234 . . 3 class {A, B}
64, 5cpr 3234 . 2 class {{A}, {A, B}}
73, 6wceq 1592 1 wff <.A, B>. = {{A}, {A, B}}
Colors of variables: wff set class
This definition is referenced by:  opeq1 3343  opeq2 3344  hbop 3352  opid 3354  opprc1 3357  opprc2 3358  opex 3699  elop 3700  opi1 3701  opi2 3702  opth 3704  opeqsn 3721  opeqpr 3722  uniop 3727  op1stb 4021  xpsspw 4253  xpsspwOLD 4254  relop 4278  dmsnsnsn 4522  funopg 4605  opwf 6223  rankopb 6265  rankop 6271  tskop 6740  gruop 6769
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