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Theorem opth 4974
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 and 𝐷 are not required to be sets due our specific ordered pair definition. (Contributed by NM, 28-May-1995.)
Hypotheses
Ref Expression
opth1.1 𝐴 ∈ V
opth1.2 𝐵 ∈ V
Assertion
Ref Expression
opth (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Proof of Theorem opth
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 opth1.1 . . . 4 𝐴 ∈ V
2 opth1.2 . . . 4 𝐵 ∈ V
31, 2opth1 4973 . . 3 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐴 = 𝐶)
41, 2opi1 4966 . . . . . . 7 {𝐴} ∈ ⟨𝐴, 𝐵
5 id 22 . . . . . . 7 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
64, 5syl5eleq 2736 . . . . . 6 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐴} ∈ ⟨𝐶, 𝐷⟩)
7 oprcl 4459 . . . . . 6 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → (𝐶 ∈ V ∧ 𝐷 ∈ V))
86, 7syl 17 . . . . 5 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → (𝐶 ∈ V ∧ 𝐷 ∈ V))
98simprd 478 . . . 4 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐷 ∈ V)
103opeq1d 4439 . . . . . . . 8 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐵⟩)
1110, 5eqtr3d 2687 . . . . . . 7 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
128simpld 474 . . . . . . . 8 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐶 ∈ V)
13 dfopg 4431 . . . . . . . 8 ((𝐶 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐶, 𝐵⟩ = {{𝐶}, {𝐶, 𝐵}})
1412, 2, 13sylancl 695 . . . . . . 7 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐵⟩ = {{𝐶}, {𝐶, 𝐵}})
1511, 14eqtr3d 2687 . . . . . 6 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐵}})
16 dfopg 4431 . . . . . . 7 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐷}})
178, 16syl 17 . . . . . 6 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐷}})
1815, 17eqtr3d 2687 . . . . 5 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {{𝐶}, {𝐶, 𝐵}} = {{𝐶}, {𝐶, 𝐷}})
19 prex 4939 . . . . . 6 {𝐶, 𝐵} ∈ V
20 prex 4939 . . . . . 6 {𝐶, 𝐷} ∈ V
2119, 20preqr2 4412 . . . . 5 ({{𝐶}, {𝐶, 𝐵}} = {{𝐶}, {𝐶, 𝐷}} → {𝐶, 𝐵} = {𝐶, 𝐷})
2218, 21syl 17 . . . 4 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐶, 𝐵} = {𝐶, 𝐷})
23 preq2 4301 . . . . . . 7 (𝑥 = 𝐷 → {𝐶, 𝑥} = {𝐶, 𝐷})
2423eqeq2d 2661 . . . . . 6 (𝑥 = 𝐷 → ({𝐶, 𝐵} = {𝐶, 𝑥} ↔ {𝐶, 𝐵} = {𝐶, 𝐷}))
25 eqeq2 2662 . . . . . 6 (𝑥 = 𝐷 → (𝐵 = 𝑥𝐵 = 𝐷))
2624, 25imbi12d 333 . . . . 5 (𝑥 = 𝐷 → (({𝐶, 𝐵} = {𝐶, 𝑥} → 𝐵 = 𝑥) ↔ ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷)))
27 vex 3234 . . . . . 6 𝑥 ∈ V
282, 27preqr2 4412 . . . . 5 ({𝐶, 𝐵} = {𝐶, 𝑥} → 𝐵 = 𝑥)
2926, 28vtoclg 3297 . . . 4 (𝐷 ∈ V → ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷))
309, 22, 29sylc 65 . . 3 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐵 = 𝐷)
313, 30jca 553 . 2 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → (𝐴 = 𝐶𝐵 = 𝐷))
32 opeq12 4435 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
3331, 32impbii 199 1 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  {csn 4210  {cpr 4212  cop 4216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217
This theorem is referenced by:  opthg  4975  otth2  4981  copsexg  4985  copsex4g  4988  opcom  4994  moop2  4995  propssopi  5000  opelopabsbALT  5013  ralxpf  5301  cnvcnvsn  5648  funopg  5960  funsndifnop  6456  tpres  6507  oprabv  6745  xpopth  7251  eqop  7252  opiota  7273  soxp  7335  fnwelem  7337  xpdom2  8096  xpf1o  8163  unxpdomlem2  8206  unxpdomlem3  8207  xpwdomg  8531  fseqenlem1  8885  iundom2g  9400  eqresr  9996  cnref1o  11865  hashfun  13262  fsumcom2  14549  fsumcom2OLD  14550  fprodcom2  14758  fprodcom2OLD  14759  qredeu  15419  qnumdenbi  15499  crth  15530  prmreclem3  15669  imasaddfnlem  16235  dprd2da  18487  dprd2d2  18489  ucnima  22132  numclwlk1lem2f1  27347  br8d  29548  xppreima2  29578  aciunf1lem  29590  ofpreima  29593  erdszelem9  31307  msubff1  31579  mvhf1  31582  brtp  31765  br8  31772  br6  31773  br4  31774  brsegle  32340  poimirlem4  33543  poimirlem9  33548  f1opr  33649  dib1dim  36771  diclspsn  36800  dihopelvalcpre  36854  dihmeetlem4preN  36912  dihmeetlem13N  36925  dih1dimatlem  36935  dihatlat  36940  pellexlem3  37712  pellex  37716  snhesn  38397  opelopab4  39084
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