Theorem opth 2782

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 2781	. . 3 |- (<.A, B>. = <.C, D>. -> A = C)
3		eqeq1 1478	. . . . 5 |- (<.A, B>. = <.C, D>. -> (<.A, B>. = <.C, B>. <-> <.C, D>. = <.C, B>.))
4		opeq1 2483	. . . . 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 2412	. . . . . . 7 |- <.C, D>. = {{C}, {C, D}}
7		df-op 2412	. . . . . . 7 |- <.C, B>. = {{C}, {C, B}}
8	6, 7	eqeq12i 1485	. . . . . 6 |- (<.C, D>. = <.C, B>. <-> {{C}, {C, D}} = {{C}, {C, B}})
9		prex 2776	. . . . . . 7 |- {C, D} e. V
10		prex 2776	. . . . . . 7 |- {C, B} e. V
11	9, 10	preqr2 2478	. . . . . 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 2478	. . . . . 6 |- ({C, D} = {C, B} -> D = B)
16	15	eqcomd 1477	. . . . 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 2485	. 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 954   e. wcel 956  Vcvv 1807  {csn 2405  {cpr 2406  <.cop 2407

This theorem is referenced by:  opthg 2783  eqvinop 2786  copsexg 2787  opth2 2795  opabid 2805  opelxp 3209  ralxpf 3215  cnvsn 3441  funopg 3539  fsn 3825  xpopth 4096  xpdom2 4428  aceq5lem4 4718  unidom 4788  eqresr 5235  ltresr 5238  xpnnen 7449  ipfval 8299

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412

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