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Definition df-op 3415
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 3600 and opprc2 3601). For Kuratowski's actual definition when the arguments are sets, see dfop 3577.

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 3415 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3415 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
StepHypRef Expression
1 cA . . 3 class 𝐴
2 cB . . 3 class 𝐵
31, 2cop 3409 . 2 class 𝐴, 𝐵
4 cvv 2602 . . . . 5 class V
51, 4wcel 1434 . . . 4 wff 𝐴 ∈ V
62, 4wcel 1434 . . . 4 wff 𝐵 ∈ V
7 vx . . . . . 6 setvar 𝑥
87cv 1284 . . . . 5 class 𝑥
91csn 3406 . . . . . 6 class {𝐴}
101, 2cpr 3407 . . . . . 6 class {𝐴, 𝐵}
119, 10cpr 3407 . . . . 5 class {{𝐴}, {𝐴, 𝐵}}
128, 11wcel 1434 . . . 4 wff 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}
135, 6, 12w3a 920 . . 3 wff (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})
1413, 7cab 2068 . 2 class {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
153, 14wceq 1285 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  3576  opeq1  3578  opeq2  3579  nfop  3594  opprc  3599  oprcl  3602  opm  3997
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