Theorem opth 2787

Description: The ordered pair theorem. If two ordered pairs are equal, their first elements are equal and their second elements are equal. Exercise 6 of [TakeutiZaring] p. 16. Note that C is not required to be a set due to a peculiarity of our specific ordered pair definition.

Hypotheses

Ref	Expression
opth.1	|- A e. V
opth.2	|- B e. V
opth.3	|- D e. V

Assertion

Ref	Expression
opth	|- (<.A, B>. = <.C, D>. <-> (A = C /\ B = D))

Proof of Theorem opth

Step	Hyp	Ref	Expression
1		opth.1	. . . 4 |- A e. V
2	1	opth1 2786	. . 3 |- (<.A, B>. = <.C, D>. -> A = C)
3		eqeq1 1481	. . . . 5 |- (<.A, B>. = <.C, D>. -> (<.A, B>. = <.C, B>. <-> <.C, D>. = <.C, B>.))
4		opeq1 2487	. . . . 5 |- (A = C -> <.A, B>. = <.C, B>.)
5	3, 4	syl5bi 208	. . . 4 |- (<.A, B>. = <.C, D>. -> (A = C -> <.C, D>. = <.C, B>.))
6		df-op 2416	. . . . . . 7 |- <.C, D>. = {{C}, {C, D}}
7		df-op 2416	. . . . . . 7 |- <.C, B>. = {{C}, {C, B}}
8	6, 7	eqeq12i 1488	. . . . . 6 |- (<.C, D>. = <.C, B>. <-> {{C}, {C, D}} = {{C}, {C, B}})
9		prex 2781	. . . . . . 7 |- {C, D} e. V
10		prex 2781	. . . . . . 7 |- {C, B} e. V
11	9, 10	preqr2 2482	. . . . . 6 |- ({{C}, {C, D}} = {{C}, {C, B}} -> {C, D} = {C, B})
12	8, 11	sylbi 199	. . . . 5 |- (<.C, D>. = <.C, B>. -> {C, D} = {C, B})
13		opth.3	. . . . . . 7 |- D e. V
14		opth.2	. . . . . . 7 |- B e. V
15	13, 14	preqr2 2482	. . . . . 6 |- ({C, D} = {C, B} -> D = B)
16	15	eqcomd 1480	. . . . 5 |- ({C, D} = {C, B} -> B = D)
17	12, 16	syl 10	. . . 4 |- (<.C, D>. = <.C, B>. -> B = D)
18	5, 17	syl6 22	. . 3 |- (<.A, B>. = <.C, D>. -> (A = C -> B = D))
19	2, 18	jcai 289	. 2 |- (<.A, B>. = <.C, D>. -> (A = C /\ B = D))
20		opeq12 2489	. 2 |- ((A = C /\ B = D) -> <.A, B>. = <.C, D>.)
21	19, 20	impbi 157	1 |- (<.A, B>. = <.C, D>. <-> (A = C /\ B = D))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811  {csn 2409  {cpr 2410  <.cop 2411

This theorem is referenced by:  opthg 2788  eqvinop 2791  copsexg 2792  opth2 2800  opabid 2810  opelxp 3214  ralxpf 3220  cnvsn 3449  funopg 3547  fsn 3834  xpopth 4106  xpdom2 4442  aceq5lem4 4738  unidom 4808  eqresr 5255  ltresr 5258  xpnnen 7499  ipfval 8352

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416

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