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Definition df-op 3540
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 3734 and opprc2 3735). For Kuratowski's actual definition when the arguments are sets, see dfop 3711.

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 3540 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3540 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, 𝐵2 = {{{𝐴}, ∅}, {{𝐵}}}. This was simplified by Kazimierz Kuratowski in 1921 to our present definition. An even simpler definition is 𝐴, 𝐵3 = {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
StepHypRef Expression
1 cA . . 3 class 𝐴
2 cB . . 3 class 𝐵
31, 2cop 3534 . 2 class 𝐴, 𝐵
4 cvv 2689 . . . . 5 class V
51, 4wcel 1481 . . . 4 wff 𝐴 ∈ V
62, 4wcel 1481 . . . 4 wff 𝐵 ∈ V
7 vx . . . . . 6 setvar 𝑥
87cv 1331 . . . . 5 class 𝑥
91csn 3531 . . . . . 6 class {𝐴}
101, 2cpr 3532 . . . . . 6 class {𝐴, 𝐵}
119, 10cpr 3532 . . . . 5 class {{𝐴}, {𝐴, 𝐵}}
128, 11wcel 1481 . . . 4 wff 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}
135, 6, 12w3a 963 . . 3 wff (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})
1413, 7cab 2126 . 2 class {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
153, 14wceq 1332 1 wff 𝐴, 𝐵⟩ = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
Colors of variables: wff set class
This definition is referenced by:  dfopg  3710  opeq1  3712  opeq2  3713  nfop  3728  opprc  3733  oprcl  3736  opm  4163
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