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 , opprc2 , and 0nelop ). For Kuratowski's actual definition when the arguments are sets, see dfop . For the justifying theorem (for sets) see opth . See dfopif for an equivalent formulation using the if operation.
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 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 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 } } } , justified by opthwiener . This was simplified by Kazimierz Kuratowski in 1921 to our present definition. An even simpler definition <. A , B >._3 = { A , { A , B } } is justified by opthreg , but it requires the Axiom of Regularity for its justification and is not commonly used. A definition that also works for proper classes is <. A , B >._4 = ( ( A X. { (/) } ) u. ( B X. { { (/) } } ) ) , justified by opthprc . Nearly at the same time as Norbert Wiener, Felix Hausdorff proposed the following definition in "Grundzüge der Mengenlehre" ("Basics of Set Theory"), p. 32, in 1914: <. A , B >._5 = { { A , O } , { B , T } } . Hausdorff used 1 and 2 instead of O and T , but actually any two different fixed sets will do (e.g., O = (/) and T = { (/) } , see 0nep0 ). Furthermore, Hausdorff demanded that O and T are both different from A as well as B , which is actually not necessary (at least not in full extent), see opthhausdorff0 and opthhausdorff . If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi . An ordered pair of real numbers can also be represented by a complex number as shown by cru . Kuratowski's ordered pair definition is standard for ZFC set theory, but it is very inconvenient to use in New Foundations theory because it is not type-level; a common alternate definition in New Foundations is the definition from Rosser p. 281.
Since there are other ways to define ordered pairs, we discourage direct use of this definition so that most theorems won't depend on this particular construction; theorems will instead rely on dfopif . (Contributed by NM, 28-May-1995) (Revised by Mario Carneiro, 26-Apr-2015) (Avoid depending on this detail.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | df-op | ⊢ ⟨ 𝐴 , 𝐵 ⟩ = { 𝑥 ∣ ( 𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ { { 𝐴 } , { 𝐴 , 𝐵 } } ) } |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 0 | cA | ⊢ 𝐴 | |
| 1 | cB | ⊢ 𝐵 | |
| 2 | 0 1 | cop | ⊢ ⟨ 𝐴 , 𝐵 ⟩ |
| 3 | vx | ⊢ 𝑥 | |
| 4 | cvv | ⊢ V | |
| 5 | 0 4 | wcel | ⊢ 𝐴 ∈ V |
| 6 | 1 4 | wcel | ⊢ 𝐵 ∈ V |
| 7 | 3 | cv | ⊢ 𝑥 |
| 8 | 0 | csn | ⊢ { 𝐴 } |
| 9 | 0 1 | cpr | ⊢ { 𝐴 , 𝐵 } |
| 10 | 8 9 | cpr | ⊢ { { 𝐴 } , { 𝐴 , 𝐵 } } |
| 11 | 7 10 | wcel | ⊢ 𝑥 ∈ { { 𝐴 } , { 𝐴 , 𝐵 } } |
| 12 | 5 6 11 | w3a | ⊢ ( 𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ { { 𝐴 } , { 𝐴 , 𝐵 } } ) |
| 13 | 12 3 | cab | ⊢ { 𝑥 ∣ ( 𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ { { 𝐴 } , { 𝐴 , 𝐵 } } ) } |
| 14 | 2 13 | wceq | ⊢ ⟨ 𝐴 , 𝐵 ⟩ = { 𝑥 ∣ ( 𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ { { 𝐴 } , { 𝐴 , 𝐵 } } ) } |