Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-op Unicode version

Definition df-op 4036
 Description: Definition of an ordered pair, equivalent to Kuratowski's definition when the arguments are sets. Since the behavior of Kuratowski definition is not very useful for proper classes, we define it to be empty in this case (see opprc1 4240, opprc2 4241, and 0nelop 4742). For Kuratowski's actual definition when the arguments are sets, see dfop 4216. For the justifying theorem (for sets) see opth 4726. See dfopif 4214 for an equivalent formulation using the if operation. Definition 9.1 of [Quine] p. 58 defines an ordered pair unconditionally as , which has different behavior from our df-op 4036 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 4036 was chosen because it often makes proofs shorter by eliminating unnecessary sethood hypotheses. 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 ={{{A}, },{{ }}}, justified by opthwiener 4754. This was simplified by Kazimierz Kuratowski in 1921 to our present definition. An even simpler definition _3 ={A,{A, }} is justified by opthreg 8056, 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 =((AX.{ })u.( X.{{ }})), justified by opthprc 5052. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 12350. An ordered pair of real numbers can also be represented by a complex number as shown by cru 10553. Kuratowski's ordered pair definition is standard for ZFC set theory, but it is very inconvenient to use in New Foundations theory because it is not type-level; a common alternative definition in New Foundations is the Definition from [Rosser] p. 281. Since there are other ways to define ordered pairs, we discourage direct use of this definition so that most theorems won't depend on this particular construction; theorems will instead rely on dfopif 4214. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.)
Assertion
Ref Expression
df-op
Distinct variable groups:   ,   ,

Detailed syntax breakdown of Definition df-op
StepHypRef Expression
1 cA . . 3
2 cB . . 3
31, 2cop 4035 . 2
4 cvv 3109 . . . . 5
51, 4wcel 1818 . . . 4
62, 4wcel 1818 . . . 4
7 vx . . . . . 6
87cv 1394 . . . . 5
91csn 4029 . . . . . 6
101, 2cpr 4031 . . . . . 6
119, 10cpr 4031 . . . . 5
128, 11wcel 1818 . . . 4
135, 6, 12w3a 973 . . 3
1413, 7cab 2442 . 2
153, 14wceq 1395 1
 Colors of variables: wff setvar class This definition is referenced by:  dfopif  4214
 Copyright terms: Public domain W3C validator