Theorem opth 5460

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 5459	. . 3 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐴 = 𝐶)
4	1, 2	opi1 5452	. . . . . . 7 ⊢ {𝐴} ∈ ⟨𝐴, 𝐵⟩
5		id 23	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
6	4, 5	eleqtrid 2871	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐴} ∈ ⟨𝐶, 𝐷⟩)
7		oprcl 4866	. . . . . 6 ⊢ ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → (𝐶 ∈ V ∧ 𝐷 ∈ V))
8	6, 7	syl 18	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → (𝐶 ∈ V ∧ 𝐷 ∈ V))
9	8	simprd 501	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐷 ∈ V)
10	3	opeq1d 4846	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐵⟩)
11	10, 5	eqtr3d 2802	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
12	8	simpld 500	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐶 ∈ V)
13		dfopg 4838	. . . . . . . 8 ⊢ ((𝐶 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐶, 𝐵⟩ = {{𝐶}, {𝐶, 𝐵}})
14	12, 2, 13	sylancl 598	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐵⟩ = {{𝐶}, {𝐶, 𝐵}})
15	11, 14	eqtr3d 2802	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐵}})
16		dfopg 4838	. . . . . . 7 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐷}})
17	8, 16	syl 18	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐷}})
18	15, 17	eqtr3d 2802	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {{𝐶}, {𝐶, 𝐵}} = {{𝐶}, {𝐶, 𝐷}})
19		prex 5411	. . . . . 6 ⊢ {𝐶, 𝐵} ∈ V
20		prex 5411	. . . . . 6 ⊢ {𝐶, 𝐷} ∈ V
21	19, 20	preqr2 4816	. . . . 5 ⊢ ({{𝐶}, {𝐶, 𝐵}} = {{𝐶}, {𝐶, 𝐷}} → {𝐶, 𝐵} = {𝐶, 𝐷})
22	18, 21	syl 18	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐶, 𝐵} = {𝐶, 𝐷})
23		preq2 4702	. . . . . . 7 ⊢ (𝑥 = 𝐷 → {𝐶, 𝑥} = {𝐶, 𝐷})
24	23	eqeq2d 2776	. . . . . 6 ⊢ (𝑥 = 𝐷 → ({𝐶, 𝐵} = {𝐶, 𝑥} ↔ {𝐶, 𝐵} = {𝐶, 𝐷}))
25		eqeq2 2777	. . . . . 6 ⊢ (𝑥 = 𝐷 → (𝐵 = 𝑥 ↔ 𝐵 = 𝐷))
26	24, 25	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝐷 → (({𝐶, 𝐵} = {𝐶, 𝑥} → 𝐵 = 𝑥) ↔ ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷)))
27		vex 3461	. . . . . 6 ⊢ 𝑥 ∈ V
28	2, 27	preqr2 4816	. . . . 5 ⊢ ({𝐶, 𝐵} = {𝐶, 𝑥} → 𝐵 = 𝑥)
29	26, 28	vtoclg 3524	. . . 4 ⊢ (𝐷 ∈ V → ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷))
30	9, 22, 29	sylc 66	. . 3 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐵 = 𝐷)
31	3, 30	jca 521	. 2 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
32		opeq12 4842	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
33	31, 32	impbii 212	1 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Vcvv 3457  {csn 4591  {cpr 4593  ⟨cop 4597

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598

This theorem is used by:  opthg  5461  otth2  5467  copsexgw  5474  copsexgwOLD  5475  copsexg  5476  copsex2g  5478  copsex4g  5480  opcom  5486  moop2  5487  propssopi  5493  brtp  5509  vopelopabsb  5515  brab2d  5524  ralxpf  5834  cnvopab  6139  cnvcnvsn  6222  opreu2reurex  6299  funopg  6574  funsndifnop  7154  tpres  7206  f1opr  7475  oprabv  7479  xpopth  8033  eqop  8034  opiota  8062  soxp  8131  fnwelem  8133  xpdom2  9067  xpf1o  9134  unxpdomlem2  9224  unxpdomlem3  9225  xpwdomg  9554  djulf1o  9914  djurf1o  9915  fseqenlem1  10024  iundom2g  10541  eqresr  11139  cnref1o  13027  hashfun  14494  fsumcom2  15850  fprodcom2  16063  qredeu  16740  qnumdenbi  16827  crth  16861  prmreclem3  17002  imasaddfnlem  17606  fnpr2ob  17636  dprd2da  20160  dprd2d2  20162  rngqiprngimf1  21492  ucnima  24490  numclwwlk1lem2f1  30781  br8d  33026  xppreima2  33069  aciunf1lem  33080  ofpreima  33083  erdszelem9  35730  goeleq12bg  35880  gonanegoal  35883  gonan0  35923  goaln0  35924  gonarlem  35925  gonar  35926  goalrlem  35927  goalr  35928  fmla0disjsuc  35929  fmlasucdisj  35930  satffunlem  35932  satffunlem1lem1  35933  satffunlem2lem1  35935  msubff1  36087  mvhf1  36090  br8  36287  br6  36288  br4  36289  brsegle  36639  nmulprop  36721  copsex2gd  37841  copsex2b  37843  poimirlem4  38334  poimirlem9  38339  dib1dim  41999  diclspsn  42028  dihopelvalcpre  42082  dihmeetlem4preN  42140  dihmeetlem13N  42153  dih1dimatlem  42163  dihatlat  42168  pellexlem3  43618  pellex  43622  snhesn  44572  opelopab4  45320  ichnreuop  48281  ichreuopeq  48282  gpgedg2ov  48891  gpgedg2iv  48892  pgnioedg1  48933  pgnioedg2  48934  pgnioedg3  48935  pgnioedg4  48936  pgnioedg5  48937  pgnbgreunbgrlem2lem1  48939  pgnbgreunbgrlem2lem2  48940  pgnbgreunbgrlem2lem3  48941  pgnbgreunbgrlem5lem1  48945  pgnbgreunbgrlem5lem2  48946  pgnbgreunbgrlem5lem3  48947  rrx2xpref1o  49557  brab2dd  49665  idfudiag1  50362