Definition df-op 3590

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 3785 and opprc2 3786). For Kuratowski's actual definition when the arguments are sets, see dfop 3762.

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 3590 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3590 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 3584	. 2 class ⟨𝐴, 𝐵⟩
4		cvv 2730	. . . . 5 class V
5	1, 4	wcel 2141	. . . 4 wff 𝐴 ∈ V
6	2, 4	wcel 2141	. . . 4 wff 𝐵 ∈ V
7		vx	. . . . . 6 setvar 𝑥
8	7	cv 1347	. . . . 5 class 𝑥
9	1	csn 3581	. . . . . 6 class {𝐴}
10	1, 2	cpr 3582	. . . . . 6 class {𝐴, 𝐵}
11	9, 10	cpr 3582	. . . . 5 class {{𝐴}, {𝐴, 𝐵}}
12	8, 11	wcel 2141	. . . 4 wff 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}
13	5, 6, 12	w3a 973	. . 3 wff (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})
14	13, 7	cab 2156	. 2 class {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
15	3, 14	wceq 1348	1 wff ⟨𝐴, 𝐵⟩ = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}

This definition is referenced by:  dfopg  3761  opeq1  3763  opeq2  3764  nfop  3779  opprc  3784  oprcl  3787  opm  4217