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Definition df-op 3087
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 3210, opprc1b 3565, opprc2 3211, and opprc3 3566). For the justifying theorem (for sets) see opth 3553. 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 3577, 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 6042, 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 4061. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 8751. Finally, an ordered pair of real numbers can be represented by a complex number as shown by crui 7689.
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 3081 . 2 class <.A, B>.
41csn 3078 . . 3 class {A}
51, 2cpr 3079 . . 3 class {A, B}
64, 5cpr 3079 . 2 class {{A}, {A, B}}
73, 6wceq 1434 1 wff <.A, B>. = {{A}, {A, B}}
Colors of variables: wff set class
This definition is referenced by:  opeq1 3196  opeq2 3197  hbop 3205  opid 3207  opprc1 3210  opprc2 3211  opex 3548  elop 3549  opi1 3550  opi2 3551  opth 3553  opeqsn 3572  opeqpr 3573  uniop 3578  op1stb 3879  xpsspw 4114  xpsspwOLD 4115  relop 4139  funopg 4468  opwf 6143  rankopb 6182  rankop 6188  tskop 6647  gruop 6681
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