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Definition df-op 3411
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 3598 and opprc2 3599). For Kuratowski's actual definition when the arguments are sets, see dfop 3575.

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 3411 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3411 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 3405 . 2  class  <. A ,  B >.
4 cvv 2574 . . . . 5  class  _V
51, 4wcel 1409 . . . 4  wff  A  e. 
_V
62, 4wcel 1409 . . . 4  wff  B  e. 
_V
7 vx . . . . . 6  setvar  x
87cv 1258 . . . . 5  class  x
91csn 3402 . . . . . 6  class  { A }
101, 2cpr 3403 . . . . . 6  class  { A ,  B }
119, 10cpr 3403 . . . . 5  class  { { A } ,  { A ,  B } }
128, 11wcel 1409 . . . 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 2042 . 2  class  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }
153, 14wceq 1259 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  3574  opeq1  3576  opeq2  3577  nfop  3592  opprc  3597  oprcl  3600  opm  3998
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