ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-op Structured version   GIF version

Definition df-op 3349
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 3535 and opprc2 3536). For Kuratowski's actual definition when the arguments are sets, see dfop 3512.

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 3349 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3349 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 V B V x {{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 3343 . 2 class A, B
4 cvv 2527 . . . . 5 class V
51, 4wcel 1367 . . . 4 wff A V
62, 4wcel 1367 . . . 4 wff B V
7 vx . . . . . 6 setvar x
87cv 1223 . . . . 5 class x
91csn 3340 . . . . . 6 class {A}
101, 2cpr 3341 . . . . . 6 class {A, B}
119, 10cpr 3341 . . . . 5 class {{A}, {A, B}}
128, 11wcel 1367 . . . 4 wff x {{A}, {A, B}}
135, 6, 12w3a 867 . . 3 wff (A V B V x {{A}, {A, B}})
1413, 7cab 2000 . 2 class {x ∣ (A V B V x {{A}, {A, B}})}
153, 14wceq 1224 1 wff A, B⟩ = {x ∣ (A V B V x {{A}, {A, B}})}
Colors of variables: wff set class
This definition is referenced by:  dfopg  3511  opeq1  3513  opeq2  3514  nfop  3529  opprc  3534  oprcl  3537  opm  3935
  Copyright terms: Public domain W3C validator