Theorem opth 3965

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 3964	. . 3 ⊢ (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → A = 𝐶)
4	1, 2	opi1 3960	. . . . . . 7 ⊢ {A} ∈ ⟨A, B⟩
5		id 19	. . . . . . 7 ⊢ (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ⟨A, B⟩ = ⟨𝐶, 𝐷⟩)
6	4, 5	syl5eleq 2123	. . . . . 6 ⊢ (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → {A} ∈ ⟨𝐶, 𝐷⟩)
7		oprcl 3564	. . . . . 6 ⊢ ({A} ∈ ⟨𝐶, 𝐷⟩ → (𝐶 ∈ V ∧ 𝐷 ∈ V))
8	6, 7	syl 14	. . . . 5 ⊢ (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → (𝐶 ∈ V ∧ 𝐷 ∈ V))
9	8	simprd 107	. . . 4 ⊢ (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → 𝐷 ∈ V)
10	3	opeq1d 3546	. . . . . . . 8 ⊢ (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ⟨A, B⟩ = ⟨𝐶, B⟩)
11	10, 5	eqtr3d 2071	. . . . . . 7 ⊢ (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, B⟩ = ⟨𝐶, 𝐷⟩)
12	8	simpld 105	. . . . . . . 8 ⊢ (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → 𝐶 ∈ V)
13		dfopg 3538	. . . . . . . 8 ⊢ ((𝐶 ∈ V ∧ B ∈ V) → ⟨𝐶, B⟩ = {{𝐶}, {𝐶, B}})
14	12, 2, 13	sylancl 392	. . . . . . 7 ⊢ (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, B⟩ = {{𝐶}, {𝐶, B}})
15	11, 14	eqtr3d 2071	. . . . . 6 ⊢ (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, B}})
16		dfopg 3538	. . . . . . 7 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐷}})
17	8, 16	syl 14	. . . . . 6 ⊢ (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐷}})
18	15, 17	eqtr3d 2071	. . . . 5 ⊢ (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → {{𝐶}, {𝐶, B}} = {{𝐶}, {𝐶, 𝐷}})
19		prexgOLD 3937	. . . . . . 7 ⊢ ((𝐶 ∈ V ∧ B ∈ V) → {𝐶, B} ∈ V)
20	12, 2, 19	sylancl 392	. . . . . 6 ⊢ (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → {𝐶, B} ∈ V)
21		prexgOLD 3937	. . . . . . 7 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → {𝐶, 𝐷} ∈ V)
22	8, 21	syl 14	. . . . . 6 ⊢ (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → {𝐶, 𝐷} ∈ V)
23		preqr2g 3529	. . . . . 6 ⊢ (({𝐶, B} ∈ V ∧ {𝐶, 𝐷} ∈ V) → ({{𝐶}, {𝐶, B}} = {{𝐶}, {𝐶, 𝐷}} → {𝐶, B} = {𝐶, 𝐷}))
24	20, 22, 23	syl2anc 391	. . . . 5 ⊢ (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ({{𝐶}, {𝐶, B}} = {{𝐶}, {𝐶, 𝐷}} → {𝐶, B} = {𝐶, 𝐷}))
25	18, 24	mpd 13	. . . 4 ⊢ (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → {𝐶, B} = {𝐶, 𝐷})
26		preq2 3439	. . . . . . 7 ⊢ (x = 𝐷 → {𝐶, x} = {𝐶, 𝐷})
27	26	eqeq2d 2048	. . . . . 6 ⊢ (x = 𝐷 → ({𝐶, B} = {𝐶, x} ↔ {𝐶, B} = {𝐶, 𝐷}))
28		eqeq2 2046	. . . . . 6 ⊢ (x = 𝐷 → (B = x ↔ B = 𝐷))
29	27, 28	imbi12d 223	. . . . 5 ⊢ (x = 𝐷 → (({𝐶, B} = {𝐶, x} → B = x) ↔ ({𝐶, B} = {𝐶, 𝐷} → B = 𝐷)))
30		vex 2554	. . . . . 6 ⊢ x ∈ V
31	2, 30	preqr2 3531	. . . . 5 ⊢ ({𝐶, B} = {𝐶, x} → B = x)
32	29, 31	vtoclg 2607	. . . 4 ⊢ (𝐷 ∈ V → ({𝐶, B} = {𝐶, 𝐷} → B = 𝐷))
33	9, 25, 32	sylc 56	. . 3 ⊢ (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → B = 𝐷)
34	3, 33	jca 290	. 2 ⊢ (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → (A = 𝐶 ∧ B = 𝐷))
35		opeq12 3542	. 2 ⊢ ((A = 𝐶 ∧ B = 𝐷) → ⟨A, B⟩ = ⟨𝐶, 𝐷⟩)
36	34, 35	impbii 117	1 ⊢ (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ ↔ (A = 𝐶 ∧ B = 𝐷))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  Vcvv 2551  {csn 3367  {cpr 3368  ⟨cop 3370

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376

This theorem is referenced by:  opthg  3966  otth2  3969  copsexg  3972  copsex4g  3975  opcom  3978  moop2  3979  opelopabsbALT  3987  opelopabsb  3988  ralxpf  4425  rexxpf  4426  cnvcnvsn  4740  funopg  4877  brabvv  5493  xpdom2  6241  enq0ref  6415  enq0tr  6416  mulnnnq0  6432  eqresr  6693  cnref1o  8317