ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-op GIF version

Definition df-op 3592
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 3787 and opprc2 3788). For Kuratowski's actual definition when the arguments are sets, see dfop 3764.

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 3592 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3592 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 3586 . 2 class 𝐴, 𝐵
4 cvv 2730 . . . . 5 class V
51, 4wcel 2141 . . . 4 wff 𝐴 ∈ V
62, 4wcel 2141 . . . 4 wff 𝐵 ∈ V
7 vx . . . . . 6 setvar 𝑥
87cv 1347 . . . . 5 class 𝑥
91csn 3583 . . . . . 6 class {𝐴}
101, 2cpr 3584 . . . . . 6 class {𝐴, 𝐵}
119, 10cpr 3584 . . . . 5 class {{𝐴}, {𝐴, 𝐵}}
128, 11wcel 2141 . . . 4 wff 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}
135, 6, 12w3a 973 . . 3 wff (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})
1413, 7cab 2156 . 2 class {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
153, 14wceq 1348 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  3763  opeq1  3765  opeq2  3766  nfop  3781  opprc  3786  oprcl  3789  opm  4219
  Copyright terms: Public domain W3C validator