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Definition df-op 3847
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 4030, opprc2 4031, and 0nelop 4475). For Kuratowski's actual definition when the arguments are sets, see dfop 4007. For the justifying theorem (for sets) see opth 4464. See dfopif 4005 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 3847 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3847 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 4487. 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 7602, 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 4954. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 11594. Finally, an ordered pair of real numbers can be represented by a complex number as shown by cru 10023. (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 3841 . 2  class  <. A ,  B >.
4 cvv 2962 . . . . 5  class  _V
51, 4wcel 1727 . . . 4  wff  A  e. 
_V
62, 4wcel 1727 . . . 4  wff  B  e. 
_V
7 vx . . . . . 6  set  x
87cv 1652 . . . . 5  class  x
91csn 3838 . . . . . 6  class  { A }
101, 2cpr 3839 . . . . . 6  class  { A ,  B }
119, 10cpr 3839 . . . . 5  class  { { A } ,  { A ,  B } }
128, 11wcel 1727 . . . 4  wff  x  e. 
{ { A } ,  { A ,  B } }
135, 6, 12w3a 937 . . 3  wff  ( A  e.  _V  /\  B  e.  _V  /\  x  e. 
{ { A } ,  { A ,  B } } )
1413, 7cab 2428 . 2  class  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }
153, 14wceq 1653 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  4005
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