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

Definition df-op 3083
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 3206, opprc1b 3561, opprc2 3207, and opprc3 3562). For the justifying theorem (for sets) see opth 3549. 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 _2 , justified by opthwiener 3573, which was simplified by Kazimierz Kuratowski in 1921 to our present definition. An even simpler definition _3 is justified by opthreg 6056, but it requires the Axiom of Regularity for its justification and is not commonly used. A definition that also works for proper classes is _4 , justified by opthprc 4057. 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 7703.
Assertion
Ref Expression
df-op

Detailed syntax breakdown of Definition df-op
StepHypRef Expression
1 cA . . 3
2 cB . . 3
31, 2cop 3077 . 2
41csn 3074 . . 3
51, 2cpr 3075 . . 3
64, 5cpr 3075 . 2
73, 6wceq 1425 1
Colors of variables: wff set class
This definition is referenced by:  opeq1 3192  opeq2 3193  hbop 3201  opid 3203  opprc1 3206  opprc2 3207  opex 3544  elop 3545  opi1 3546  opi2 3547  opth 3549  opeqsn 3568  opeqpr 3569  uniop 3574  op1stb 3875  xpsspw 4110  xpsspwOLD 4111  relop 4135  funopg 4468  opwf 6157  rankopb 6196  rankop 6202  tskop 6661  gruop 6695
Copyright terms: Public domain