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Definition df-op 3504
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 3695 and opprc2 3696). For Kuratowski's actual definition when the arguments are sets, see dfop 3672.

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 3504 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3504 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 } } }. This was simplified by Kazimierz Kuratowski in 1921 to our present definition. An even simpler definition is  <. A ,  B >.3  =  { A ,  { A ,  B } }, but it requires the Axiom of Regularity for its justification and is not commonly used. Finally, an ordered pair of real numbers can be represented by a complex number. (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 3498 . 2  class  <. A ,  B >.
4 cvv 2658 . . . . 5  class  _V
51, 4wcel 1463 . . . 4  wff  A  e. 
_V
62, 4wcel 1463 . . . 4  wff  B  e. 
_V
7 vx . . . . . 6  setvar  x
87cv 1313 . . . . 5  class  x
91csn 3495 . . . . . 6  class  { A }
101, 2cpr 3496 . . . . . 6  class  { A ,  B }
119, 10cpr 3496 . . . . 5  class  { { A } ,  { A ,  B } }
128, 11wcel 1463 . . . 4  wff  x  e. 
{ { A } ,  { A ,  B } }
135, 6, 12w3a 945 . . 3  wff  ( A  e.  _V  /\  B  e.  _V  /\  x  e. 
{ { A } ,  { A ,  B } } )
1413, 7cab 2101 . 2  class  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }
153, 14wceq 1314 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:  dfopg  3671  opeq1  3673  opeq2  3674  nfop  3689  opprc  3694  oprcl  3697  opm  4124
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