Definition df-op 3601

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 3800 and opprc2 3801). For Kuratowski's actual definition when the arguments are sets, see dfop 3777.

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 3601 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3601 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 3595	. 2 class ⟨𝐴, 𝐵⟩
4		cvv 2737	. . . . 5 class V
5	1, 4	wcel 2148	. . . 4 wff 𝐴 ∈ V
6	2, 4	wcel 2148	. . . 4 wff 𝐵 ∈ V
7		vx	. . . . . 6 setvar 𝑥
8	7	cv 1352	. . . . 5 class 𝑥
9	1	csn 3592	. . . . . 6 class {𝐴}
10	1, 2	cpr 3593	. . . . . 6 class {𝐴, 𝐵}
11	9, 10	cpr 3593	. . . . 5 class {{𝐴}, {𝐴, 𝐵}}
12	8, 11	wcel 2148	. . . 4 wff 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}
13	5, 6, 12	w3a 978	. . 3 wff (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})
14	13, 7	cab 2163	. 2 class {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
15	3, 14	wceq 1353	1 wff ⟨𝐴, 𝐵⟩ = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}

This definition is referenced by:  dfopg  3776  opeq1  3778  opeq2  3779  nfop  3794  opprc  3799  oprcl  3802  opm  4234