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Theorem opkelsikg 4264
 Description: Membership in Kuratowski singleton image. (Contributed by SF, 13-Jan-2015.)
Assertion
Ref Expression
opkelsikg ((A V B W) → (⟪A, B SIk Cxy(A = {x} B = {y} x, y C)))
Distinct variable groups:   x,A,y   x,B,y   x,C,y
Allowed substitution hints:   V(x,y)   W(x,y)

Proof of Theorem opkelsikg
Dummy variables z t u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sik 4192 . 2 SIk C = {z tu(z = ⟪t, u xy(t = {x} u = {y} x, y C))}
2 eqeq1 2359 . . . 4 (t = A → (t = {x} ↔ A = {x}))
323anbi1d 1256 . . 3 (t = A → ((t = {x} u = {y} x, y C) ↔ (A = {x} u = {y} x, y C)))
432exbidv 1628 . 2 (t = A → (xy(t = {x} u = {y} x, y C) ↔ xy(A = {x} u = {y} x, y C)))
5 eqeq1 2359 . . . 4 (u = B → (u = {y} ↔ B = {y}))
653anbi2d 1257 . . 3 (u = B → ((A = {x} u = {y} x, y C) ↔ (A = {x} B = {y} x, y C)))
762exbidv 1628 . 2 (u = B → (xy(A = {x} u = {y} x, y C) ↔ xy(A = {x} B = {y} x, y C)))
81, 4, 7opkelopkabg 4245 1 ((A V B W) → (⟪A, B SIk Cxy(A = {x} B = {y} x, 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   SIk csik 4181 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-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-sik 4192 This theorem is referenced by:  opksnelsik  4265  dfpw12  4301  setconslem1  4731  dfsi2  4751
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