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Theorem opth 4726
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 and are not required to be sets due our specific ordered pair definition. (Contributed by NM, 28-May-1995.)
Hypotheses
Ref Expression
opth1.1
opth1.2
Assertion
Ref Expression
opth

Proof of Theorem opth
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 opth1.1 . . . 4
2 opth1.2 . . . 4
31, 2opth1 4725 . . 3
41, 2opi1 4719 . . . . . . 7
5 id 22 . . . . . . 7
64, 5syl5eleq 2551 . . . . . 6
7 oprcl 4242 . . . . . 6
86, 7syl 16 . . . . 5
98simprd 463 . . . 4
103opeq1d 4223 . . . . . . . 8
1110, 5eqtr3d 2500 . . . . . . 7
128simpld 459 . . . . . . . 8
13 dfopg 4215 . . . . . . . 8
1412, 2, 13sylancl 662 . . . . . . 7
1511, 14eqtr3d 2500 . . . . . 6
16 dfopg 4215 . . . . . . 7
178, 16syl 16 . . . . . 6
1815, 17eqtr3d 2500 . . . . 5
19 prex 4694 . . . . . 6
20 prex 4694 . . . . . 6
2119, 20preqr2 4205 . . . . 5
2218, 21syl 16 . . . 4
23 preq2 4110 . . . . . . 7
2423eqeq2d 2471 . . . . . 6
25 eqeq2 2472 . . . . . 6
2624, 25imbi12d 320 . . . . 5
27 vex 3112 . . . . . 6
282, 27preqr2 4205 . . . . 5
2926, 28vtoclg 3167 . . . 4
309, 22, 29sylc 60 . . 3
313, 30jca 532 . 2
32 opeq12 4219 . 2
3331, 32impbii 188 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818   cvv 3109  {csn 4029  {cpr 4031  <.cop 4035
This theorem is referenced by:  opthg  4727  otth2  4733  copsexg  4737  copsexgOLD  4738  copsex4g  4741  opcom  4746  moop2  4747  opelopabsbALT  4761  ralxpf  5154  cnvcnvsn  5490  funopg  5625  oprabv  6345  xpopth  6839  eqop  6840  opiota  6859  soxp  6913  fnwelem  6915  xpdom2  7632  xpf1o  7699  unxpdomlem2  7745  unxpdomlem3  7746  xpwdomg  8032  fseqenlem1  8426  iundom2g  8936  eqresr  9535  cnref1o  11244  hashfun  12495  fsumcom2  13589  fprodcom2  13788  xpnnenOLD  13943  qredeu  14248  qnumdenbi  14277  crt  14308  prmreclem3  14436  imasaddfnlem  14925  dprd2da  17091  dprd2d2  17093  ucnima  20784  numclwlk1lem2f1  25094  br8d  27463  xppreima2  27488  ofpreima  27507  erdszelem9  28643  msubff1  28916  mvhf1  28919  brtp  29178  br8  29185  br6  29186  br4  29187  brsegle  29758  f1opr  30215  pellexlem3  30767  pellex  30771  tpres  32554  opelopab4  33324  dib1dim  36892  diclspsn  36921  dihopelvalcpre  36975  dihmeetlem4preN  37033  dihmeetlem13N  37046  dih1dimatlem  37056  dihatlat  37061  snhesn  37809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036
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