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Theorem elopab 3986
Description: Membership in a class abstraction of pairs. (Contributed by NM, 24-Mar-1998.)
Assertion
Ref Expression
elopab (A {⟨x, y⟩ ∣ φ} ↔ xy(A = ⟨x, y φ))
Distinct variable groups:   x,A   y,A
Allowed substitution hints:   φ(x,y)

Proof of Theorem elopab
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 elex 2560 . 2 (A {⟨x, y⟩ ∣ φ} → A V)
2 vex 2554 . . . . . 6 x V
3 vex 2554 . . . . . 6 y V
42, 3opex 3957 . . . . 5 x, y V
5 eleq1 2097 . . . . 5 (A = ⟨x, y⟩ → (A V ↔ ⟨x, y V))
64, 5mpbiri 157 . . . 4 (A = ⟨x, y⟩ → A V)
76adantr 261 . . 3 ((A = ⟨x, y φ) → A V)
87exlimivv 1773 . 2 (xy(A = ⟨x, y φ) → A V)
9 eqeq1 2043 . . . . 5 (z = A → (z = ⟨x, y⟩ ↔ A = ⟨x, y⟩))
109anbi1d 438 . . . 4 (z = A → ((z = ⟨x, y φ) ↔ (A = ⟨x, y φ)))
11102exbidv 1745 . . 3 (z = A → (xy(z = ⟨x, y φ) ↔ xy(A = ⟨x, y φ)))
12 df-opab 3810 . . 3 {⟨x, y⟩ ∣ φ} = {zxy(z = ⟨x, y φ)}
1311, 12elab2g 2683 . 2 (A V → (A {⟨x, y⟩ ∣ φ} ↔ xy(A = ⟨x, y φ)))
141, 8, 13pm5.21nii 619 1 (A {⟨x, y⟩ ∣ φ} ↔ xy(A = ⟨x, y φ))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551  cop 3370  {copab 3808
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  df-opab 3810
This theorem is referenced by:  opelopabsbALT  3987  opelopabsb  3988  opelopabt  3990  opelopabga  3991  opabm  4008  iunopab  4009  epelg  4018  elxp  4305  elcnv  4455  dfmpt3  4964  0neqopab  5492  brabvv  5493  opabex3d  5690  opabex3  5691
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