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Theorem opkelcok 4262
 Description: Membership in a Kuratowski composition. (Contributed by SF, 13-Jan-2015.)
Hypotheses
Ref Expression
opkelcok.1 A V
opkelcok.2 B V
Assertion
Ref Expression
opkelcok (⟪A, B (C k D) ↔ x(⟪A, x D x, B C))
Distinct variable groups:   x,A   x,B   x,C   x,D

Proof of Theorem opkelcok
StepHypRef Expression
1 opkelcok.1 . 2 A V
2 opkelcok.2 . 2 B V
3 opkelcokg 4261 . 2 ((A V B V) → (⟪A, B (C k D) ↔ x(⟪A, x D x, B C)))
41, 2, 3mp2an 653 1 (⟪A, B (C k D) ↔ x(⟪A, x D x, B C))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   ∈ wcel 1710  Vcvv 2859  ⟪copk 4057   ∘k ccomk 4180 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-rex 2620  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-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190 This theorem is referenced by:  imacok  4282  dfnnc2  4395  nnsucelrlem1  4424  dfop2lem1  4573  setconslem1  4731  setconslem2  4732  dfco1  4748
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