Definition df-op 3278

Description: Kuratowski's ordered pair definition. Definition 9.1 of [Quine] p. 58. For proper classes it is not meaningful but is well-defined and we allow it for convenience (see opprc1 3391, opprc1b 3737, opprc2 3392, and opprc3 3738). For the justifying theorem (for sets) see opth 3727. 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 3749, which 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 5985, 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 4211. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 8416. Finally, an ordered pair of real numbers can be represented by a complex number as shown by crui 8487.

Assertion

Ref	Expression
df-op	|- <.A, B>. = {{A}, {A, B}}

Detailed syntax breakdown of Definition df-op

Step	Hyp	Ref	Expression
1		cA	. . 3 class A
2		cB	. . 3 class B
3	1, 2	cop 3272	. 2 class <.A, B>.
4	1	csn 3270	. . 3 class {A}
5	1, 2	cpr 3271	. . 3 class {A, B}
6	4, 5	cpr 3271	. 2 class {{A}, {A, B}}
7	3, 6	wceq 1615	1 wff <.A, B>. = {{A}, {A, B}}

This definition is referenced by:  opeq1 3377  opeq2 3378  hbop 3386  opid 3388  opprc1 3391  opprc2 3392  opex 3722  elop 3723  opi1 3724  opi2 3725  opth 3727  opeqsn 3744  opeqpr 3745  uniop 3750  op1stb 4035  xpsspw 4255  relop 4274  dmsnsnsn 4513  funopg 4594  rankop 6105  orkurssOLD 15447  tarorpa 16307

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