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Definition df-op 4002
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 4219, opprc2 4220, and 0nelop 4732). For Kuratowski's actual definition when the arguments are sets, see dfop 4195. For the justifying theorem (for sets) see opth 4717. See dfopif 4193 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 4002 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 4002 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 4744. 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 8208, 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 × {∅}) ∪ (𝐵 × {{∅}})), justified by opthprc 4929. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 12570. An ordered pair of real numbers can also be represented by a complex number as shown by cru 10690. Kuratowski's ordered pair definition is standard for ZFC set theory, but it is very inconvenient to use in New Foundations theory because it is not type-level; a common alternate definition in New Foundations is the definition from [Rosser] p. 281.

Since there are other ways to define ordered pairs, we discourage direct use of this definition so that most theorems won't depend on this particular construction; theorems will instead rely on dfopif 4193. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.)

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 4001 . 2 class 𝐴, 𝐵
4 cvv 3066 . . . . 5 class V
51, 4wcel 1937 . . . 4 wff 𝐴 ∈ V
62, 4wcel 1937 . . . 4 wff 𝐵 ∈ V
7 vx . . . . . 6 setvar 𝑥
87cv 1467 . . . . 5 class 𝑥
91csn 3995 . . . . . 6 class {𝐴}
101, 2cpr 3997 . . . . . 6 class {𝐴, 𝐵}
119, 10cpr 3997 . . . . 5 class {{𝐴}, {𝐴, 𝐵}}
128, 11wcel 1937 . . . 4 wff 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}
135, 6, 12w3a 1021 . . 3 wff (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})
1413, 7cab 2491 . 2 class {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
153, 14wceq 1468 1 wff 𝐴, 𝐵⟩ = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
Colors of variables: wff setvar class
This definition is referenced by:  dfopif  4193
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