Theorem opth 5458

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 5457	. . 3 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐴 = 𝐶)
4	1, 2	opi1 5450	. . . . . . 7 ⊢ {𝐴} ∈ ⟨𝐴, 𝐵⟩
5		id 23	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
6	4, 5	eleqtrid 2869	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐴} ∈ ⟨𝐶, 𝐷⟩)
7		oprcl 4864	. . . . . 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 500	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐷 ∈ V)
10	3	opeq1d 4844	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐵⟩)
11	10, 5	eqtr3d 2800	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
12	8	simpld 499	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐶 ∈ V)
13		dfopg 4836	. . . . . . . 8 ⊢ ((𝐶 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐶, 𝐵⟩ = {{𝐶}, {𝐶, 𝐵}})
14	12, 2, 13	sylancl 597	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐵⟩ = {{𝐶}, {𝐶, 𝐵}})
15	11, 14	eqtr3d 2800	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐵}})
16		dfopg 4836	. . . . . . 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 2800	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {{𝐶}, {𝐶, 𝐵}} = {{𝐶}, {𝐶, 𝐷}})
19		prex 5409	. . . . . 6 ⊢ {𝐶, 𝐵} ∈ V
20		prex 5409	. . . . . 6 ⊢ {𝐶, 𝐷} ∈ V
21	19, 20	preqr2 4814	. . . . 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 4700	. . . . . . 7 ⊢ (𝑥 = 𝐷 → {𝐶, 𝑥} = {𝐶, 𝐷})
24	23	eqeq2d 2774	. . . . . 6 ⊢ (𝑥 = 𝐷 → ({𝐶, 𝐵} = {𝐶, 𝑥} ↔ {𝐶, 𝐵} = {𝐶, 𝐷}))
25		eqeq2 2775	. . . . . 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 3459	. . . . . 6 ⊢ 𝑥 ∈ V
28	2, 27	preqr2 4814	. . . . 5 ⊢ ({𝐶, 𝐵} = {𝐶, 𝑥} → 𝐵 = 𝑥)
29	26, 28	vtoclg 3522	. . . 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 520	. 2 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
32		opeq12 4840	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
33	31, 32	impbii 212	1 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455  {csn 4589  {cpr 4591  ⟨cop 4595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596

This theorem is referenced by:  opthg  5459  otth2  5465  copsexgw  5472  copsexgwOLD  5473  copsexg  5474  copsex2g  5476  copsex4g  5478  opcom  5484  moop2  5485  propssopi  5491  brtp  5507  vopelopabsb  5513  brab2d  5522  ralxpf  5832  cnvopab  6137  cnvcnvsn  6220  opreu2reurex  6295  funopg  6570  funsndifnop  7148  tpres  7199  f1opr  7466  oprabv  7470  xpopth  8023  eqop  8024  opiota  8052  soxp  8121  fnwelem  8123  xpdom2  9056  xpf1o  9123  unxpdomlem2  9213  unxpdomlem3  9214  xpwdomg  9543  djulf1o  9894  djurf1o  9895  fseqenlem1  10004  iundom2g  10519  eqresr  11117  cnref1o  13004  hashfun  14470  fsumcom2  15821  fprodcom2  16034  qredeu  16711  qnumdenbi  16798  crth  16832  prmreclem3  16973  imasaddfnlem  17577  fnpr2ob  17607  dprd2da  20109  dprd2d2  20111  rngqiprngimf1  21440  ucnima  24437  numclwwlk1lem2f1  30708  br8d  32953  xppreima2  32996  aciunf1lem  33007  ofpreima  33010  erdszelem9  35691  goeleq12bg  35841  gonanegoal  35844  gonan0  35884  goaln0  35885  gonarlem  35886  gonar  35887  goalrlem  35888  goalr  35889  fmla0disjsuc  35890  fmlasucdisj  35891  satffunlem  35893  satffunlem1lem1  35894  satffunlem2lem1  35896  msubff1  36048  mvhf1  36051  br8  36248  br6  36249  br4  36250  brsegle  36600  nmulprop  36682  copsex2gd  37802  copsex2b  37804  poimirlem4  38295  poimirlem9  38300  dib1dim  41959  diclspsn  41988  dihopelvalcpre  42042  dihmeetlem4preN  42100  dihmeetlem13N  42113  dih1dimatlem  42123  dihatlat  42128  pellexlem3  43578  pellex  43582  snhesn  44532  opelopab4  45280  ichnreuop  48241  ichreuopeq  48242  gpgedg2ov  48851  gpgedg2iv  48852  pgnioedg1  48893  pgnioedg2  48894  pgnioedg3  48895  pgnioedg4  48896  pgnioedg5  48897  pgnbgreunbgrlem2lem1  48899  pgnbgreunbgrlem2lem2  48900  pgnbgreunbgrlem2lem3  48901  pgnbgreunbgrlem5lem1  48905  pgnbgreunbgrlem5lem2  48906  pgnbgreunbgrlem5lem3  48907  rrx2xpref1o  49518  brab2dd  49626  idfudiag1  50323