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Definition df-op 3376
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 3562 and opprc2 3563). For Kuratowski's actual definition when the arguments are sets, see dfop 3539.

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 3376 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3376 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 3370 . 2 class A, B
4 cvv 2551 . . . . 5 class V
51, 4wcel 1390 . . . 4 wff A V
62, 4wcel 1390 . . . 4 wff B V
7 vx . . . . . 6 setvar x
87cv 1241 . . . . 5 class x
91csn 3367 . . . . . 6 class {A}
101, 2cpr 3368 . . . . . 6 class {A, B}
119, 10cpr 3368 . . . . 5 class {{A}, {A, B}}
128, 11wcel 1390 . . . 4 wff x {{A}, {A, B}}
135, 6, 12w3a 884 . . 3 wff (A V B V x {{A}, {A, B}})
1413, 7cab 2023 . 2 class {x ∣ (A V B V x {{A}, {A, B}})}
153, 14wceq 1242 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  3538  opeq1  3540  opeq2  3541  nfop  3556  opprc  3561  oprcl  3564  opm  3962
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