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Definition df-op 3281
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 3430, opprc2 3431, and 0nelop 3824). For Kuratowski's actual definition when the arguments are sets, see dfop 3411. For the justifying theorem (for sets) see opth 3813. See dfopif 3409 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 3281 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3281 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 3836. 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 6782, 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 4338. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 10209. Finally, an ordered pair of real numbers can be represented by a complex number as shown by crui 8853. (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 3275 . 2  class  <. A ,  B >.
4 cvv 2492 . . . . 5  class  _V
51, 4wcel 1538 . . . 4  wff  A  e.  _V
62, 4wcel 1538 . . . 4  wff  B  e.  _V
7 vx . . . . . 6  set  x
87cv 1535 . . . . 5  class  x
91csn 3272 . . . . . 6  class  { A }
101, 2cpr 3273 . . . . . 6  class  { A ,  B }
119, 10cpr 3273 . . . . 5  class  { { A } ,  { A ,  B } }
128, 11wcel 1538 . . . 4  wff  x  e.  { { A } ,  { A ,  B } }
135, 6, 12w3a 903 . . 3  wff  ( A  e.  _V  /\  B  e. 
_V  /\  x  e.  { { A } ,  { A ,  B } } )
1413, 7cab 2069 . 2  class  { x  |  ( A  e.  _V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }
153, 14wceq 1536 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  3409
Copyright terms: Public domain