ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  opth Structured version   GIF version

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.
StepHypRef Expression
1 opth1.1 . . . 4 A V
2 opth1.2 . . . 4 B V
31, 2opth1 3947 . . 3 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → A = 𝐶)
41, 2opi1 3943 . . . . . . 7 {A} A, B
5 id 19 . . . . . . 7 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ⟨A, B⟩ = ⟨𝐶, 𝐷⟩)
64, 5syl5eleq 2108 . . . . . 6 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → {A} 𝐶, 𝐷⟩)
7 oprcl 3547 . . . . . 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 3529 . . . . . . . 8 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ⟨A, B⟩ = ⟨𝐶, B⟩)
1110, 5eqtr3d 2056 . . . . . . 7 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, B⟩ = ⟨𝐶, 𝐷⟩)
128simpld 105 . . . . . . . 8 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → 𝐶 V)
13 dfopg 3521 . . . . . . . 8 ((𝐶 V B V) → ⟨𝐶, B⟩ = {{𝐶}, {𝐶, B}})
1412, 2, 13sylancl 394 . . . . . . 7 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, B⟩ = {{𝐶}, {𝐶, B}})
1511, 14eqtr3d 2056 . . . . . 6 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, B}})
16 dfopg 3521 . . . . . . 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 2056 . . . . 5 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → {{𝐶}, {𝐶, B}} = {{𝐶}, {𝐶, 𝐷}})
19 prexgOLD 3920 . . . . . . 7 ((𝐶 V B V) → {𝐶, B} V)
2012, 2, 19sylancl 394 . . . . . 6 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → {𝐶, B} V)
21 prexgOLD 3920 . . . . . . 7 ((𝐶 V 𝐷 V) → {𝐶, 𝐷} V)
228, 21syl 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}))
2420, 22, 23syl2anc 393 . . . . 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 3422 . . . . . . 7 (x = 𝐷 → {𝐶, x} = {𝐶, 𝐷})
2726eqeq2d 2033 . . . . . 6 (x = 𝐷 → ({𝐶, B} = {𝐶, x} ↔ {𝐶, B} = {𝐶, 𝐷}))
28 eqeq2 2031 . . . . . 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 2538 . . . . . 6 x V
312, 30preqr2 3514 . . . . 5 ({𝐶, B} = {𝐶, x} → B = x)
3229, 31vtoclg 2590 . . . 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 3525 . 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 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
  Copyright terms: Public domain W3C validator