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Definition df-op 3303
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 3459, opprc2 3460, and 0nelop 3854). For Kuratowski's actual definition when the arguments are sets, see dfop 3440. For the justifying theorem (for sets) see opth 3843. See dfopif 3438 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 3303 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3303 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 3866. 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 6846, 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 4363. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 10692. Finally, an ordered pair of real numbers can be represented by a complex number as shown by cru 9163. (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 3297 . 2  class  <. A ,  B >.
4 cvv 2512 . . . . 5  class  _V
51, 4wcel 1533 . . . 4  wff  A  e. 
_V
62, 4wcel 1533 . . . 4  wff  B  e. 
_V
7 vx . . . . . 6  set  x
87cv 1530 . . . . 5  class  x
91csn 3294 . . . . . 6  class  { A }
101, 2cpr 3295 . . . . . 6  class  { A ,  B }
119, 10cpr 3295 . . . . 5  class  { { A } ,  { A ,  B } }
128, 11wcel 1533 . . . 4  wff  x  e. 
{ { A } ,  { A ,  B } }
135, 6, 12w3a 900 . . 3  wff  ( A  e.  _V  /\  B  e.  _V  /\  x  e. 
{ { A } ,  { A ,  B } } )
1413, 7cab 2087 . 2  class  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }
153, 14wceq 1531 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  3438
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