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Theorem opkabssvvki 4209
 Description: Any Kuratowski ordered pair abstraction is a subset of (V ×k V). (Contributed by SF, 13-Jan-2015.)
Hypothesis
Ref Expression
opkabssvvki.1 A = {x yz(x = ⟪y, z φ)}
Assertion
Ref Expression
opkabssvvki A (V ×k V)
Distinct variable groups:   x,y   x,z
Allowed substitution hints:   φ(x,y,z)   A(x,y,z)

Proof of Theorem opkabssvvki
StepHypRef Expression
1 opkabssvvki.1 . 2 A = {x yz(x = ⟪y, z φ)}
2 opkabssvvk 4208 . 2 {x yz(x = ⟪y, z φ)} (V ×k V)
31, 2eqsstri 3301 1 A (V ×k V)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642  {cab 2339  Vcvv 2859   ⊆ wss 3257  ⟪copk 4057   ×k cxpk 4174 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-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-xpk 4185 This theorem is referenced by:  xpkssvvk  4210  sikssvvk  4266  cnvkssvvk  4275  ssetkssvvk  4278  ins2kss  4279  ins3kss  4280  idkssvvk  4281
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