Definition df-op 4184

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 4425, opprc2 4426, and 0nelop 4960). For Kuratowski's actual definition when the arguments are sets, see dfop 4401. For the justifying theorem (for sets) see opth 4945. See dfopif 4399 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 4184 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 4184 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 4976. 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 8515, 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 5167. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 13057. An ordered pair of real numbers can also be represented by a complex number as shown by cru 11012. 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 4399. (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 4183	. 2 class ⟨𝐴, 𝐵⟩
4		cvv 3200	. . . . 5 class V
5	1, 4	wcel 1990	. . . 4 wff 𝐴 ∈ V
6	2, 4	wcel 1990	. . . 4 wff 𝐵 ∈ V
7		vx	. . . . . 6 setvar 𝑥
8	7	cv 1482	. . . . 5 class 𝑥
9	1	csn 4177	. . . . . 6 class {𝐴}
10	1, 2	cpr 4179	. . . . . 6 class {𝐴, 𝐵}
11	9, 10	cpr 4179	. . . . 5 class {{𝐴}, {𝐴, 𝐵}}
12	8, 11	wcel 1990	. . . 4 wff 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}
13	5, 6, 12	w3a 1037	. . 3 wff (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})
14	13, 7	cab 2608	. 2 class {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
15	3, 14	wceq 1483	1 wff ⟨𝐴, 𝐵⟩ = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}

This definition is referenced by:  dfopif  4399