NFE Home New Foundations Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  NFE Home  >  Th. List  >  elopab GIF version

Theorem elopab 4696
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 2867 . 2 (A {x, y φ} → A V)
2 vex 2862 . . . . . 6 x V
3 vex 2862 . . . . . 6 y V
42, 3opex 4588 . . . . 5 x, y V
5 eleq1 2413 . . . . 5 (A = x, y → (A V ↔ x, y V))
64, 5mpbiri 224 . . . 4 (A = x, yA V)
76adantr 451 . . 3 ((A = x, y φ) → A V)
87exlimivv 1635 . 2 (xy(A = x, y φ) → A V)
9 eqeq1 2359 . . . . 5 (z = A → (z = x, yA = x, y))
109anbi1d 685 . . . 4 (z = A → ((z = x, y φ) ↔ (A = x, y φ)))
11102exbidv 1628 . . 3 (z = A → (xy(z = x, y φ) ↔ xy(A = x, y φ)))
12 df-opab 4623 . . 3 {x, y φ} = {z xy(z = x, y φ)}
1311, 12elab2g 2987 . 2 (A V → (A {x, y φ} ↔ xy(A = x, y φ)))
141, 8, 13pm5.21nii 342 1 (A {x, y φ} ↔ xy(A = x, y φ))
Colors of variables: wff setvar class
Syntax hints:  wb 176   wa 358  wex 1541   = wceq 1642   wcel 1710  Vcvv 2859  cop 4561  {copab 4622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-opab 4623
This theorem is referenced by:  opelopabt  4699  opelopabga  4700  opabn0  4716  el1st  4729  setconslem4  4734  elswap  4740  elxp  4801  elcnv  4889  composeex  5820
  Copyright terms: Public domain W3C validator