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Definition df-op 3100
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 3221, opprc1b 3574, opprc2 3222, and opprc3 3575). For the justifying theorem (for sets) see opth 3564. 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 3586, 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 6049, 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 4070. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 8756. Finally, an ordered pair of real numbers can be represented by a complex number as shown by crui 7696.
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 3094 . 2 class <.A, B>.
41csn 3092 . . 3 class {A}
51, 2cpr 3093 . . 3 class {A, B}
64, 5cpr 3093 . 2 class {{A}, {A, B}}
73, 6wceq 1449 1 wff <.A, B>. = {{A}, {A, B}}
Colors of variables: wff set class
This definition is referenced by:  opeq1 3207  opeq2 3208  hbop 3216  opid 3218  opprc1 3221  opprc2 3222  opex 3559  elop 3560  opi1 3561  opi2 3562  opth 3564  opeqsn 3581  opeqpr 3582  uniop 3587  op1stb 3888  xpsspw 4123  xpsspwOLD 4124  relop 4148  funopg 4477  opwf 6151  rankopb 6190  rankop 6196  tskop 6655  gruop 6689
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