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Definition df-op 3355
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 3541 and opprc2 3542). For Kuratowski's actual definition when the arguments are sets, see dfop 3518.

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 3355 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3355 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 3349 . 2 class A, B
4 cvv 2531 . . . . 5 class V
51, 4wcel 1370 . . . 4 wff A V
62, 4wcel 1370 . . . 4 wff B V
7 vx . . . . . 6 setvar x
87cv 1225 . . . . 5 class x
91csn 3346 . . . . . 6 class {A}
101, 2cpr 3347 . . . . . 6 class {A, B}
119, 10cpr 3347 . . . . 5 class {{A}, {A, B}}
128, 11wcel 1370 . . . 4 wff x {{A}, {A, B}}
135, 6, 12w3a 871 . . 3 wff (A V B V x {{A}, {A, B}})
1413, 7cab 2004 . 2 class {x ∣ (A V B V x {{A}, {A, B}})}
153, 14wceq 1226 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  3517  opeq1  3519  opeq2  3520  nfop  3535  opprc  3540  oprcl  3543  opm  3941
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