Theorem opth 3532

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 3531	. . 3 |- (<.A, B>. = <.C, D>. -> A = C)
3		eqeq1 1890	. . . . 5 |- (<.A, B>. = <.C, D>. -> (<.A, B>. = <.C, B>. <-> <.C, D>. = <.C, B>.))
4		opeq1 3158	. . . . 5 |- (A = C -> <.A, B>. = <.C, B>.)
5	3, 4	syl5bi 225	. . . 4 |- (<.A, B>. = <.C, D>. -> (A = C -> <.C, D>. = <.C, B>.))
6		df-op 3053	. . . . . . 7 |- <.C, D>. = {{C}, {C, D}}
7		df-op 3053	. . . . . . 7 |- <.C, B>. = {{C}, {C, B}}
8	6, 7	eqeq12i 1897	. . . . . 6 |- (<.C, D>. = <.C, B>. <-> {{C}, {C, D}} = {{C}, {C, B}})
9		prex 3526	. . . . . . 7 |- {C, D} e. _V
10		prex 3526	. . . . . . 7 |- {C, B} e. _V
11	9, 10	preqr2 3153	. . . . . 6 |- ({{C}, {C, D}} = {{C}, {C, B}} -> {C, D} = {C, B})
12	8, 11	sylbi 216	. . . . 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 3153	. . . . . 6 |- ({C, D} = {C, B} -> D = B)
16	15	eqcomd 1889	. . . . 5 |- ({C, D} = {C, B} -> B = D)
17	12, 16	syl 12	. . . 4 |- (<.C, D>. = <.C, B>. -> B = D)
18	5, 17	syl6 25	. . 3 |- (<.A, B>. = <.C, D>. -> (A = C -> B = D))
19	2, 18	jcai 313	. 2 |- (<.A, B>. = <.C, D>. -> (A = C /\ B = D))
20		opeq12 3160	. 2 |- ((A = C /\ B = D) -> <.A, B>. = <.C, D>.)
21	19, 20	impbii 174	1 |- (<.A, B>. = <.C, D>. <-> (A = C /\ B = D))

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292  {csn 3044  {cpr 3045  <.cop 3046

This theorem is referenced by:  opthg 3533  eqvinop 3536  copsexg 3537  copsexgOLD 3538  opth2 3546  opabidOLD 3558  opelopabsb 3564  opelxpOLD 4037  ralxpf 4043  cnvsn 4373  funopg 4454  funsn 4463  iunfopabOLD 4543  fsn 4807  xpopth 5046  xpdom2 5501  aceq5lem4 5900  unidom 5970  eqresr 6407  ltresr 6410  xpnnen 8768  ipfval 9691  bnj136 12468  brtp 13830  soxp 13950  f1opr 15714

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053

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