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Definition df-op 3260
Description: Definition of an ordered pair, equivalent to Kuratowski's definition  { { A } ,  { A ,  B } } 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 3412, opprc2 3413, and 0nelop 3806). For Kuratowski's actual definition when the arguments are sets, see dfop 3393. For the justifying theorem (for sets) see opth 3795. See dfopif 3391 for an equivalent formulation using the  if operation.

Definition 9.1 of [Quine] p. 58 defines an ordered pair unconditionally as  <. A ,  B >.  =  { { A } ,  { A ,  B } }, which has different behavior from our df-op 3260 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3260 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  <. A ,  B >._2  =  { { { A } ,  (/) } ,  { { B } } }, justified by opthwiener 3818. This 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 6792, 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 4315. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 10224. Finally, an ordered pair of real numbers can be represented by a complex number as shown by crui 8863. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion
Ref Expression
df-op  |-  <. A ,  B >.  =  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }
Distinct variable groups:    x, A    x, B

Detailed syntax breakdown of Definition df-op
StepHypRef Expression
1 cA . . 3  class  A
2 cB . . 3  class  B
31, 2cop 3254 . 2  class  <. A ,  B >.
4 cvv 2472 . . . . 5  class  _V
51, 4wcel 1519 . . . 4  wff  A  e. 
_V
62, 4wcel 1519 . . . 4  wff  B  e. 
_V
7 vx . . . . . 6  set  x
87cv 1516 . . . . 5  class  x
91csn 3251 . . . . . 6  class  { A }
101, 2cpr 3252 . . . . . 6  class  { A ,  B }
119, 10cpr 3252 . . . . 5  class  { { A } ,  { A ,  B } }
128, 11wcel 1519 . . . 4  wff  x  e. 
{ { A } ,  { A ,  B } }
135, 6, 12w3a 893 . . 3  wff  ( A  e.  _V  /\  B  e.  _V  /\  x  e. 
{ { A } ,  { A ,  B } } )
1413, 7cab 2048 . 2  class  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }
153, 14wceq 1517 1  wff  <. A ,  B >.  =  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }
Colors of variables: wff set class
This definition is referenced by:  dfopif  3391
Copyright terms: Public domain