Definition df-op 4655

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 4921, opprc2 4922, and 0nelop 5515). For Kuratowski's actual definition when the arguments are sets, see dfop 4896. For the justifying theorem (for sets) see opth 5496. See dfopif 4894 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 4655 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 4655 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 5533. 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 9687, 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 5764. Nearly at the same time as Norbert Wiener, Felix Hausdorff proposed the following definition in "Grundzüge der Mengenlehre" ("Basics of Set Theory"), p. 32, in 1914: ⟨𝐴, 𝐵⟩₅ = {{𝐴, 𝑂}, {𝐵, 𝑇}}. Hausdorff used 1 and 2 instead of 𝑂 and 𝑇, but actually any two different fixed sets will do (e.g., 𝑂 = ∅ and 𝑇 = {∅}, see 0nep0 5376). Furthermore, Hausdorff demanded that 𝑂 and 𝑇 are both different from 𝐴 as well as 𝐵, which is actually not necessary (at least not in full extent), see opthhausdorff0 5537 and opthhausdorff 5536. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 14319. An ordered pair of real numbers can also be represented by a complex number as shown by cru 12285. 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 4894. (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

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		cB	. . 3 class 𝐵
3	1, 2	cop 4654	. 2 class ⟨𝐴, 𝐵⟩
4		cvv 3488	. . . . 5 class V
5	1, 4	wcel 2108	. . . 4 wff 𝐴 ∈ V
6	2, 4	wcel 2108	. . . 4 wff 𝐵 ∈ V
7		vx	. . . . . 6 setvar 𝑥
8	7	cv 1536	. . . . 5 class 𝑥
9	1	csn 4648	. . . . . 6 class {𝐴}
10	1, 2	cpr 4650	. . . . . 6 class {𝐴, 𝐵}
11	9, 10	cpr 4650	. . . . 5 class {{𝐴}, {𝐴, 𝐵}}
12	8, 11	wcel 2108	. . . 4 wff 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}
13	5, 6, 12	w3a 1087	. . 3 wff (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})
14	13, 7	cab 2717	. 2 class {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
15	3, 14	wceq 1537	1 wff ⟨𝐴, 𝐵⟩ = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}

This definition is referenced by:  dfopif  4894