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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.
StepHypRef Expression
1 opth1.1 . . . 4 A V
2 opth1.2 . . . 4 B V
31, 2opth1 3964 . . 3 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → A = 𝐶)
41, 2opi1 3960 . . . . . . 7 {A} A, B
5 id 19 . . . . . . 7 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ⟨A, B⟩ = ⟨𝐶, 𝐷⟩)
64, 5syl5eleq 2123 . . . . . 6 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → {A} 𝐶, 𝐷⟩)
7 oprcl 3564 . . . . . 6 ({A} 𝐶, 𝐷⟩ → (𝐶 V 𝐷 V))
86, 7syl 14 . . . . 5 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → (𝐶 V 𝐷 V))
98simprd 107 . . . 4 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → 𝐷 V)
103opeq1d 3546 . . . . . . . 8 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ⟨A, B⟩ = ⟨𝐶, B⟩)
1110, 5eqtr3d 2071 . . . . . . 7 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, B⟩ = ⟨𝐶, 𝐷⟩)
128simpld 105 . . . . . . . 8 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → 𝐶 V)
13 dfopg 3538 . . . . . . . 8 ((𝐶 V B V) → ⟨𝐶, B⟩ = {{𝐶}, {𝐶, B}})
1412, 2, 13sylancl 392 . . . . . . 7 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, B⟩ = {{𝐶}, {𝐶, B}})
1511, 14eqtr3d 2071 . . . . . 6 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, B}})
16 dfopg 3538 . . . . . . 7 ((𝐶 V 𝐷 V) → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐷}})
178, 16syl 14 . . . . . 6 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐷}})
1815, 17eqtr3d 2071 . . . . 5 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → {{𝐶}, {𝐶, B}} = {{𝐶}, {𝐶, 𝐷}})
19 prexgOLD 3937 . . . . . . 7 ((𝐶 V B V) → {𝐶, B} V)
2012, 2, 19sylancl 392 . . . . . 6 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → {𝐶, B} V)
21 prexgOLD 3937 . . . . . . 7 ((𝐶 V 𝐷 V) → {𝐶, 𝐷} V)
228, 21syl 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}))
2420, 22, 23syl2anc 391 . . . . 5 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ({{𝐶}, {𝐶, B}} = {{𝐶}, {𝐶, 𝐷}} → {𝐶, B} = {𝐶, 𝐷}))
2518, 24mpd 13 . . . 4 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → {𝐶, B} = {𝐶, 𝐷})
26 preq2 3439 . . . . . . 7 (x = 𝐷 → {𝐶, x} = {𝐶, 𝐷})
2726eqeq2d 2048 . . . . . 6 (x = 𝐷 → ({𝐶, B} = {𝐶, x} ↔ {𝐶, B} = {𝐶, 𝐷}))
28 eqeq2 2046 . . . . . 6 (x = 𝐷 → (B = xB = 𝐷))
2927, 28imbi12d 223 . . . . 5 (x = 𝐷 → (({𝐶, B} = {𝐶, x} → B = x) ↔ ({𝐶, B} = {𝐶, 𝐷} → B = 𝐷)))
30 vex 2554 . . . . . 6 x V
312, 30preqr2 3531 . . . . 5 ({𝐶, B} = {𝐶, x} → B = x)
3229, 31vtoclg 2607 . . . 4 (𝐷 V → ({𝐶, B} = {𝐶, 𝐷} → B = 𝐷))
339, 25, 32sylc 56 . . 3 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → B = 𝐷)
343, 33jca 290 . 2 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → (A = 𝐶 B = 𝐷))
35 opeq12 3542 . 2 ((A = 𝐶 B = 𝐷) → ⟨A, B⟩ = ⟨𝐶, 𝐷⟩)
3634, 35impbii 117 1 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ ↔ (A = 𝐶 B = 𝐷))
Colors of variables: wff set class
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
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