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Definition df-op 3381
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 3568 and opprc2 3569). For Kuratowski's actual definition when the arguments are sets, see dfop 3545.

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 3381 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3381 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 3375 . 2 class 𝐴, 𝐵
4 cvv 2554 . . . . 5 class V
51, 4wcel 1393 . . . 4 wff 𝐴 ∈ V
62, 4wcel 1393 . . . 4 wff 𝐵 ∈ V
7 vx . . . . . 6 setvar 𝑥
87cv 1242 . . . . 5 class 𝑥
91csn 3372 . . . . . 6 class {𝐴}
101, 2cpr 3373 . . . . . 6 class {𝐴, 𝐵}
119, 10cpr 3373 . . . . 5 class {{𝐴}, {𝐴, 𝐵}}
128, 11wcel 1393 . . . 4 wff 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}
135, 6, 12w3a 885 . . 3 wff (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})
1413, 7cab 2026 . 2 class {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
153, 14wceq 1243 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  3544  opeq1  3546  opeq2  3547  nfop  3562  opprc  3567  oprcl  3570  opm  3968
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