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Definition df-op 3103
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 3224, opprc1b 3577, opprc2 3225, and opprc3 3578). For the justifying theorem (for sets) see opth 3567. 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 3589, 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 6011, 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 4066. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 8697. Finally, an ordered pair of real numbers can be represented by a complex number as shown by crui 7653.
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 3097 . 2 class <.A, B>.
41csn 3095 . . 3 class {A}
51, 2cpr 3096 . . 3 class {A, B}
64, 5cpr 3096 . 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 3210  opeq2 3211  hbop 3219  opid 3221  opprc1 3224  opprc2 3225  opex 3562  elop 3563  opi1 3564  opi2 3565  opth 3567  opeqsn 3584  opeqpr 3585  uniop 3590  op1stb 3884  xpsspw 4119  xpsspwOLD 4120  relop 4144  dmsnsnsn 4389  funopg 4472  opwf 6113  rankopb 6152  rankop 6158  tskop 6627  gruop 6656
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