Definition df-op 3627

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 3826 and opprc2 3827). For Kuratowski's actual definition when the arguments are sets, see dfop 3803.

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 3627 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3627 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

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		cB	. . 3 class 𝐵
3	1, 2	cop 3621	. 2 class ⟨𝐴, 𝐵⟩
4		cvv 2760	. . . . 5 class V
5	1, 4	wcel 2164	. . . 4 wff 𝐴 ∈ V
6	2, 4	wcel 2164	. . . 4 wff 𝐵 ∈ V
7		vx	. . . . . 6 setvar 𝑥
8	7	cv 1363	. . . . 5 class 𝑥
9	1	csn 3618	. . . . . 6 class {𝐴}
10	1, 2	cpr 3619	. . . . . 6 class {𝐴, 𝐵}
11	9, 10	cpr 3619	. . . . 5 class {{𝐴}, {𝐴, 𝐵}}
12	8, 11	wcel 2164	. . . 4 wff 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}
13	5, 6, 12	w3a 980	. . 3 wff (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})
14	13, 7	cab 2179	. 2 class {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
15	3, 14	wceq 1364	1 wff ⟨𝐴, 𝐵⟩ = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}

This definition is referenced by:  dfopg  3802  opeq1  3804  opeq2  3805  nfop  3820  opprc  3825  oprcl  3828  opm  4263