Theorem opth 5431

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 = 𝐶 ∧ 𝐵 = 𝐷))

Proof of Theorem opth

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opth1.1	. . . 4 ⊢ 𝐴 ∈ V
2		opth1.2	. . . 4 ⊢ 𝐵 ∈ V
3	1, 2	opth1 5430	. . 3 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐴 = 𝐶)
4	1, 2	opi1 5423	. . . . . . 7 ⊢ {𝐴} ∈ ⟨𝐴, 𝐵⟩
5		id 22	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
6	4, 5	eleqtrid 2834	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐴} ∈ ⟨𝐶, 𝐷⟩)
7		oprcl 4859	. . . . . 6 ⊢ ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → (𝐶 ∈ V ∧ 𝐷 ∈ V))
8	6, 7	syl 17	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → (𝐶 ∈ V ∧ 𝐷 ∈ V))
9	8	simprd 495	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐷 ∈ V)
10	3	opeq1d 4839	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐵⟩)
11	10, 5	eqtr3d 2766	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
12	8	simpld 494	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐶 ∈ V)
13		dfopg 4831	. . . . . . . 8 ⊢ ((𝐶 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐶, 𝐵⟩ = {{𝐶}, {𝐶, 𝐵}})
14	12, 2, 13	sylancl 586	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐵⟩ = {{𝐶}, {𝐶, 𝐵}})
15	11, 14	eqtr3d 2766	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐵}})
16		dfopg 4831	. . . . . . 7 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐷}})
17	8, 16	syl 17	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐷}})
18	15, 17	eqtr3d 2766	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {{𝐶}, {𝐶, 𝐵}} = {{𝐶}, {𝐶, 𝐷}})
19		prex 5387	. . . . . 6 ⊢ {𝐶, 𝐵} ∈ V
20		prex 5387	. . . . . 6 ⊢ {𝐶, 𝐷} ∈ V
21	19, 20	preqr2 4809	. . . . 5 ⊢ ({{𝐶}, {𝐶, 𝐵}} = {{𝐶}, {𝐶, 𝐷}} → {𝐶, 𝐵} = {𝐶, 𝐷})
22	18, 21	syl 17	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐶, 𝐵} = {𝐶, 𝐷})
23		preq2 4694	. . . . . . 7 ⊢ (𝑥 = 𝐷 → {𝐶, 𝑥} = {𝐶, 𝐷})
24	23	eqeq2d 2740	. . . . . 6 ⊢ (𝑥 = 𝐷 → ({𝐶, 𝐵} = {𝐶, 𝑥} ↔ {𝐶, 𝐵} = {𝐶, 𝐷}))
25		eqeq2 2741	. . . . . 6 ⊢ (𝑥 = 𝐷 → (𝐵 = 𝑥 ↔ 𝐵 = 𝐷))
26	24, 25	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝐷 → (({𝐶, 𝐵} = {𝐶, 𝑥} → 𝐵 = 𝑥) ↔ ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷)))
27		vex 3448	. . . . . 6 ⊢ 𝑥 ∈ V
28	2, 27	preqr2 4809	. . . . 5 ⊢ ({𝐶, 𝐵} = {𝐶, 𝑥} → 𝐵 = 𝑥)
29	26, 28	vtoclg 3517	. . . 4 ⊢ (𝐷 ∈ V → ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷))
30	9, 22, 29	sylc 65	. . 3 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐵 = 𝐷)
31	3, 30	jca 511	. 2 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
32		opeq12 4835	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
33	31, 32	impbii 209	1 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3444  {csn 4585  {cpr 4587  ⟨cop 4591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592

This theorem is referenced by:  opthg  5432  otth2  5438  copsexgw  5445  copsexg  5446  copsex2g  5448  copsex4g  5450  opcom  5456  moop2  5457  propssopi  5463  brtp  5478  vopelopabsb  5484  ralxpf  5800  cnvopab  6098  cnvcnvsn  6180  opreu2reurex  6255  funopg  6534  funsndifnop  7105  tpres  7157  f1opr  7425  oprabv  7429  xpopth  7988  eqop  7989  opiota  8017  soxp  8085  fnwelem  8087  xpdom2  9013  xpf1o  9080  unxpdomlem2  9174  unxpdomlem3  9175  xpwdomg  9514  djulf1o  9841  djurf1o  9842  fseqenlem1  9953  iundom2g  10469  eqresr  11066  cnref1o  12920  hashfun  14378  fsumcom2  15716  fprodcom2  15926  qredeu  16604  qnumdenbi  16690  crth  16724  prmreclem3  16865  imasaddfnlem  17467  fnpr2ob  17497  dprd2da  19958  dprd2d2  19960  rngqiprngimf1  21242  ucnima  24201  numclwwlk1lem2f1  30336  brab2d  32585  br8d  32588  xppreima2  32625  aciunf1lem  32636  ofpreima  32639  erdszelem9  35179  goeleq12bg  35329  gonanegoal  35332  gonan0  35372  goaln0  35373  gonarlem  35374  gonar  35375  goalrlem  35376  goalr  35377  fmla0disjsuc  35378  fmlasucdisj  35379  satffunlem  35381  satffunlem1lem1  35382  satffunlem2lem1  35384  msubff1  35536  mvhf1  35539  br8  35736  br6  35737  br4  35738  brsegle  36089  copsex2b  37121  poimirlem4  37611  poimirlem9  37616  dib1dim  41152  diclspsn  41181  dihopelvalcpre  41235  dihmeetlem4preN  41293  dihmeetlem13N  41306  dih1dimatlem  41316  dihatlat  41321  pellexlem3  42812  pellex  42816  snhesn  43768  opelopab4  44534  ichnreuop  47466  ichreuopeq  47467  gpgedg2ov  48050  gpgedg2iv  48051  pgnioedg1  48091  pgnioedg2  48092  pgnioedg3  48093  pgnioedg4  48094  pgnioedg5  48095  pgnbgreunbgrlem2lem1  48097  pgnbgreunbgrlem2lem2  48098  pgnbgreunbgrlem2lem3  48099  pgnbgreunbgrlem5lem1  48103  pgnbgreunbgrlem5lem2  48104  pgnbgreunbgrlem5lem3  48105  rrx2xpref1o  48700  brab2dd  48809  idfudiag1  49507