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Definition df-op 3506
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 3697 and opprc2 3698). For Kuratowski's actual definition when the arguments are sets, see dfop 3674.

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 3506 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3506 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 3500 . 2  class  <. A ,  B >.
4 cvv 2660 . . . . 5  class  _V
51, 4wcel 1465 . . . 4  wff  A  e. 
_V
62, 4wcel 1465 . . . 4  wff  B  e. 
_V
7 vx . . . . . 6  setvar  x
87cv 1315 . . . . 5  class  x
91csn 3497 . . . . . 6  class  { A }
101, 2cpr 3498 . . . . . 6  class  { A ,  B }
119, 10cpr 3498 . . . . 5  class  { { A } ,  { A ,  B } }
128, 11wcel 1465 . . . 4  wff  x  e. 
{ { A } ,  { A ,  B } }
135, 6, 12w3a 947 . . 3  wff  ( A  e.  _V  /\  B  e.  _V  /\  x  e. 
{ { A } ,  { A ,  B } } )
1413, 7cab 2103 . 2  class  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }
153, 14wceq 1316 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  3673  opeq1  3675  opeq2  3676  nfop  3691  opprc  3696  oprcl  3699  opm  4126
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