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Theorem opkelidkg 4274
Description: Membership in the Kuratowski identity relationship. (Contributed by SF, 13-Jan-2015.)
Assertion
Ref Expression
opkelidkg ((A V B W) → (⟪A, B IkA = B))

Proof of Theorem opkelidkg
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-idk 4195 . 2 Ik = {z xy(z = ⟪x, y x = y)}
2 eqeq1 2359 . 2 (x = A → (x = yA = y))
3 eqeq2 2362 . 2 (y = B → (A = yA = B))
41, 2, 3opkelopkabg 4245 1 ((A V B W) → (⟪A, B IkA = B))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 176   wa 358   = wceq 1642   wcel 1710  copk 4057   Ik cidk 4184
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-idk 4195
This theorem is referenced by:  dfidk2  4313  nnsucelrlem1  4424  nndisjeq  4429  eqtfinrelk  4486  oddfinex  4504  evenodddisjlem1  4515  dfphi2  4569
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