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

Definition df-op 3224
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 3340, opprc1b 3697, opprc2 3341, and opprc3 3698). For the justifying theorem (for sets) see opth 3687. 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 3709, 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 6098, 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 4186. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 8818. Finally, an ordered pair of real numbers can be represented by a complex number as shown by crui 7917.
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 3218 . 2 class <.A, B>.
41csn 3216 . . 3 class {A}
51, 2cpr 3217 . . 3 class {A, B}
64, 5cpr 3217 . 2 class {{A}, {A, B}}
73, 6wceq 1573 1 wff <.A, B>. = {{A}, {A, B}}
Colors of variables: wff set class
This definition is referenced by:  opeq1 3326  opeq2 3327  hbop 3335  opid 3337  opprc1 3340  opprc2 3341  opex 3682  elop 3683  opi1 3684  opi2 3685  opth 3687  opeqsn 3704  opeqpr 3705  uniop 3710  op1stb 4004  xpsspw 4236  xpsspwOLD 4237  relop 4261  dmsnsnsn 4505  funopg 4588  opwf 6206  rankopb 6248  rankop 6254  tskop 6723  gruop 6752
Copyright terms: Public domain