Definition df-op 3647

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 3847 and opprc2 3848). For Kuratowski's actual definition when the arguments are sets, see dfop 3824.

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 3647 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3647 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 3641	. 2 class ⟨𝐴, 𝐵⟩
4		cvv 2773	. . . . 5 class V
5	1, 4	wcel 2177	. . . 4 wff 𝐴 ∈ V
6	2, 4	wcel 2177	. . . 4 wff 𝐵 ∈ V
7		vx	. . . . . 6 setvar 𝑥
8	7	cv 1372	. . . . 5 class 𝑥
9	1	csn 3638	. . . . . 6 class {𝐴}
10	1, 2	cpr 3639	. . . . . 6 class {𝐴, 𝐵}
11	9, 10	cpr 3639	. . . . 5 class {{𝐴}, {𝐴, 𝐵}}
12	8, 11	wcel 2177	. . . 4 wff 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}
13	5, 6, 12	w3a 981	. . 3 wff (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})
14	13, 7	cab 2192	. 2 class {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
15	3, 14	wceq 1373	1 wff ⟨𝐴, 𝐵⟩ = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}

This definition is referenced by:  dfopg  3823  opeq1  3825  opeq2  3826  nfop  3841  opprc  3846  oprcl  3849  opm  4286