HomeHome Metamath Proof Explorer < Previous   Next >
Related theorems
Unicode version

Definition df-op 3088
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 3211, opprc1b 3566, opprc2 3212, and opprc3 3567). For the justifying theorem (for sets) see opth 3554. 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 3578, 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 6057, 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 4062. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 8766. Finally, an ordered pair of real numbers can be represented by a complex number as shown by crui 7704.
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 3082 . 2 class <.A, B>.
41csn 3079 . . 3 class {A}
51, 2cpr 3080 . . 3 class {A, B}
64, 5cpr 3080 . 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 3197  opeq2 3198  hbop 3206  opid 3208  opprc1 3211  opprc2 3212  opex 3549  elop 3550  opi1 3551  opi2 3552  opth 3554  opeqsn 3573  opeqpr 3574  uniop 3579  op1stb 3880  xpsspw 4115  xpsspwOLD 4116  relop 4140  funopg 4473  opwf 6158  rankopb 6197  rankop 6203  tskop 6662  gruop 6696
Copyright terms: Public domain