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Theorem opkelins3kg 4252
 Description: Kuratowski ordered pair membership in Kuratowski insertion operator. (Contributed by SF, 12-Jan-2015.)
Assertion
Ref Expression
opkelins3kg ((A V B W) → (⟪A, B Ins3k Cxyz(A = {{x}} B = ⟪y, zx, y C)))
Distinct variable groups:   x,A,y,z   x,B,y,z   x,C,y,z
Allowed substitution hints:   V(x,y,z)   W(x,y,z)

Proof of Theorem opkelins3kg
Dummy variables w t u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ins3k 4188 . 2 Ins3k C = {t wu(t = ⟪w, u xyz(w = {{x}} u = ⟪y, zx, y C))}
2 eqeq1 2359 . . . 4 (w = A → (w = {{x}} ↔ A = {{x}}))
323anbi1d 1256 . . 3 (w = A → ((w = {{x}} u = ⟪y, zx, y C) ↔ (A = {{x}} u = ⟪y, zx, y C)))
433exbidv 1629 . 2 (w = A → (xyz(w = {{x}} u = ⟪y, zx, y C) ↔ xyz(A = {{x}} u = ⟪y, zx, y C)))
5 eqeq1 2359 . . . 4 (u = B → (u = ⟪y, z⟫ ↔ B = ⟪y, z⟫))
653anbi2d 1257 . . 3 (u = B → ((A = {{x}} u = ⟪y, zx, y C) ↔ (A = {{x}} B = ⟪y, zx, y C)))
763exbidv 1629 . 2 (u = B → (xyz(A = {{x}} u = ⟪y, zx, y C) ↔ xyz(A = {{x}} B = ⟪y, zx, y C)))
81, 4, 7opkelopkabg 4245 1 ((A V B W) → (⟪A, B Ins3k Cxyz(A = {{x}} B = ⟪y, zx, y C)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {csn 3737  ⟪copk 4057   Ins3k cins3k 4177 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-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-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-opk 4058  df-ins3k 4188 This theorem is referenced by:  otkelins3kg  4254  ins3kss  4280
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