Theorem opth 3948

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 3947	. . 3 ⊢ (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → A = 𝐶)
4	1, 2	opi1 3943	. . . . . . 7 ⊢ {A} ∈ ⟨A, B⟩
5		id 19	. . . . . . 7 ⊢ (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ⟨A, B⟩ = ⟨𝐶, 𝐷⟩)
6	4, 5	syl5eleq 2108	. . . . . 6 ⊢ (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → {A} ∈ ⟨𝐶, 𝐷⟩)
7		oprcl 3547	. . . . . 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 3529	. . . . . . . 8 ⊢ (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ⟨A, B⟩ = ⟨𝐶, B⟩)
11	10, 5	eqtr3d 2056	. . . . . . 7 ⊢ (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, B⟩ = ⟨𝐶, 𝐷⟩)
12	8	simpld 105	. . . . . . . 8 ⊢ (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → 𝐶 ∈ V)
13		dfopg 3521	. . . . . . . 8 ⊢ ((𝐶 ∈ V ∧ B ∈ V) → ⟨𝐶, B⟩ = {{𝐶}, {𝐶, B}})
14	12, 2, 13	sylancl 394	. . . . . . 7 ⊢ (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, B⟩ = {{𝐶}, {𝐶, B}})
15	11, 14	eqtr3d 2056	. . . . . 6 ⊢ (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, B}})
16		dfopg 3521	. . . . . . 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 2056	. . . . 5 ⊢ (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → {{𝐶}, {𝐶, B}} = {{𝐶}, {𝐶, 𝐷}})
19		prexgOLD 3920	. . . . . . 7 ⊢ ((𝐶 ∈ V ∧ B ∈ V) → {𝐶, B} ∈ V)
20	12, 2, 19	sylancl 394	. . . . . 6 ⊢ (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → {𝐶, B} ∈ V)
21		prexgOLD 3920	. . . . . . 7 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → {𝐶, 𝐷} ∈ V)
22	8, 21	syl 14	. . . . . 6 ⊢ (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → {𝐶, 𝐷} ∈ V)
23		preqr2g 3512	. . . . . 6 ⊢ (({𝐶, B} ∈ V ∧ {𝐶, 𝐷} ∈ V) → ({{𝐶}, {𝐶, B}} = {{𝐶}, {𝐶, 𝐷}} → {𝐶, B} = {𝐶, 𝐷}))
24	20, 22, 23	syl2anc 393	. . . . 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 3422	. . . . . . 7 ⊢ (x = 𝐷 → {𝐶, x} = {𝐶, 𝐷})
27	26	eqeq2d 2033	. . . . . 6 ⊢ (x = 𝐷 → ({𝐶, B} = {𝐶, x} ↔ {𝐶, B} = {𝐶, 𝐷}))
28		eqeq2 2031	. . . . . 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 2538	. . . . . 6 ⊢ x ∈ V
31	2, 30	preqr2 3514	. . . . 5 ⊢ ({𝐶, B} = {𝐶, x} → B = x)
32	29, 31	vtoclg 2590	. . . 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 3525	. 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 1228   ∈ wcel 1374  Vcvv 2535  {csn 3350  {cpr 3351  ⟨cop 3353

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359

This theorem is referenced by:  opthg  3949  otth2  3952  copsexg  3955  copsex4g  3958  opcom  3961  moop2  3962  opelopabsbALT  3970  opelopabsb  3971  ralxpf  4409  rexxpf  4410  cnvcnvsn  4724  funopg  4860  brabvv  5474  enq0ref  6288  enq0tr  6289  mulnnnq0  6305  eqresr  6547