MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opth Structured version   Visualization version   GIF version

Theorem opth 4935
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 4934 . . 3 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐴 = 𝐶)
41, 2opi1 4928 . . . . . . 7 {𝐴} ∈ ⟨𝐴, 𝐵
5 id 22 . . . . . . 7 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
64, 5syl5eleq 2705 . . . . . 6 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐴} ∈ ⟨𝐶, 𝐷⟩)
7 oprcl 4418 . . . . . 6 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → (𝐶 ∈ V ∧ 𝐷 ∈ V))
86, 7syl 17 . . . . 5 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → (𝐶 ∈ V ∧ 𝐷 ∈ V))
98simprd 479 . . . 4 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐷 ∈ V)
103opeq1d 4399 . . . . . . . 8 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐵⟩)
1110, 5eqtr3d 2656 . . . . . . 7 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
128simpld 475 . . . . . . . 8 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐶 ∈ V)
13 dfopg 4391 . . . . . . . 8 ((𝐶 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐶, 𝐵⟩ = {{𝐶}, {𝐶, 𝐵}})
1412, 2, 13sylancl 693 . . . . . . 7 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐵⟩ = {{𝐶}, {𝐶, 𝐵}})
1511, 14eqtr3d 2656 . . . . . 6 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐵}})
16 dfopg 4391 . . . . . . 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 2656 . . . . 5 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {{𝐶}, {𝐶, 𝐵}} = {{𝐶}, {𝐶, 𝐷}})
19 prex 4900 . . . . . 6 {𝐶, 𝐵} ∈ V
20 prex 4900 . . . . . 6 {𝐶, 𝐷} ∈ V
2119, 20preqr2 4372 . . . . 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 4260 . . . . . . 7 (𝑥 = 𝐷 → {𝐶, 𝑥} = {𝐶, 𝐷})
2423eqeq2d 2630 . . . . . 6 (𝑥 = 𝐷 → ({𝐶, 𝐵} = {𝐶, 𝑥} ↔ {𝐶, 𝐵} = {𝐶, 𝐷}))
25 eqeq2 2631 . . . . . 6 (𝑥 = 𝐷 → (𝐵 = 𝑥𝐵 = 𝐷))
2624, 25imbi12d 334 . . . . 5 (𝑥 = 𝐷 → (({𝐶, 𝐵} = {𝐶, 𝑥} → 𝐵 = 𝑥) ↔ ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷)))
27 vex 3198 . . . . . 6 𝑥 ∈ V
282, 27preqr2 4372 . . . . 5 ({𝐶, 𝐵} = {𝐶, 𝑥} → 𝐵 = 𝑥)
2926, 28vtoclg 3261 . . . 4 (𝐷 ∈ V → ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷))
309, 22, 29sylc 65 . . 3 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐵 = 𝐷)
313, 30jca 554 . 2 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → (𝐴 = 𝐶𝐵 = 𝐷))
32 opeq12 4395 . 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 384   = wceq 1481  wcel 1988  Vcvv 3195  {csn 4168  {cpr 4170  cop 4174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175
This theorem is referenced by:  opthg  4936  otth2  4942  copsexg  4946  copsex4g  4949  opcom  4955  moop2  4956  propssopi  4961  opelopabsbALT  4974  ralxpf  5257  cnvcnvsn  5600  funopg  5910  funsndifnop  6401  tpres  6451  oprabv  6688  xpopth  7192  eqop  7193  opiota  7214  soxp  7275  fnwelem  7277  xpdom2  8040  xpf1o  8107  unxpdomlem2  8150  unxpdomlem3  8151  xpwdomg  8475  fseqenlem1  8832  iundom2g  9347  eqresr  9943  cnref1o  11812  hashfun  13207  fsumcom2  14486  fsumcom2OLD  14487  fprodcom2  14695  fprodcom2OLD  14696  qredeu  15353  qnumdenbi  15433  crth  15464  prmreclem3  15603  imasaddfnlem  16169  dprd2da  18422  dprd2d2  18424  ucnima  22066  numclwlk1lem2f1  27198  br8d  29394  xppreima2  29423  aciunf1lem  29435  ofpreima  29439  erdszelem9  31155  msubff1  31427  mvhf1  31430  brtp  31614  br8  31621  br6  31622  br4  31623  brsegle  32190  poimirlem4  33384  poimirlem9  33389  f1opr  33490  dib1dim  36273  diclspsn  36302  dihopelvalcpre  36356  dihmeetlem4preN  36414  dihmeetlem13N  36427  dih1dimatlem  36437  dihatlat  36442  pellexlem3  37214  pellex  37218  snhesn  37900  opelopab4  38587
  Copyright terms: Public domain W3C validator