Definition df-op 3053

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 3170, opprc1b 3542, opprc2 3171, and opprc3 3543). For the justifying theorem (for sets) see opth 3532. 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 3554, 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 5709, 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 4046. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 7916. Finally, an ordered pair of real numbers can be represented by a complex number as shown by crui 7987.

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 3046	. 2 class <.A, B>.
4	1	csn 3044	. . 3 class {A}
5	1, 2	cpr 3045	. . 3 class {A, B}
6	4, 5	cpr 3045	. 2 class {{A}, {A, B}}
7	3, 6	wceq 1298	1 wff <.A, B>. = {{A}, {A, B}}

This definition is referenced by:  opeq1 3158  opeq2 3159  hbop 3168  opprc1 3170  opprc2 3171  opex 3527  elop 3528  opi1 3529  opi2 3530  opth 3532  opeqsn 3549  opeqpr 3550  uniop 3555  op1stb 3857  xpsspw 4093  relop 4113  dmsnsnsn 4371  funopg 4454  rankop 5804  orkurss 14488  tarorpa 15236

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