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Definition df-op 3269
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 3425, opprc2 3426, and 0nelop 3820). For Kuratowski's actual definition when the arguments are sets, see dfop 3406. For the justifying theorem (for sets) see opth 3809. See dfopif 3404 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 3269 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3269 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 3832. 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 6812, 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 4329. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 10658. Finally, an ordered pair of real numbers can be represented by a complex number as shown by cru 9129. (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 3263 . 2  class  <. A ,  B >.
4 cvv 2480 . . . . 5  class  _V
51, 4wcel 1526 . . . 4  wff  A  e. 
_V
62, 4wcel 1526 . . . 4  wff  B  e. 
_V
7 vx . . . . . 6  set  x
87cv 1523 . . . . 5  class  x
91csn 3260 . . . . . 6  class  { A }
101, 2cpr 3261 . . . . . 6  class  { A ,  B }
119, 10cpr 3261 . . . . 5  class  { { A } ,  { A ,  B } }
128, 11wcel 1526 . . . 4  wff  x  e. 
{ { A } ,  { A ,  B } }
135, 6, 12w3a 896 . . 3  wff  ( A  e.  _V  /\  B  e.  _V  /\  x  e. 
{ { A } ,  { A ,  B } } )
1413, 7cab 2056 . 2  class  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }
153, 14wceq 1524 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  3404
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