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Theorem opkelxpkg 4247
 Description: Kuratowski ordered pair membership in a Kuratowski cross product. (Contributed by SF, 12-Jan-2015.)
Assertion
Ref Expression
opkelxpkg ((A V B W) → (⟪A, B (C ×k D) ↔ (A C B D)))

Proof of Theorem opkelxpkg
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-xpk 4185 . 2 (C ×k D) = {z xy(z = ⟪x, y (x C y D))}
2 eleq1 2413 . . 3 (x = A → (x CA C))
32anbi1d 685 . 2 (x = A → ((x C y D) ↔ (A C y D)))
4 eleq1 2413 . . 3 (y = B → (y DB D))
54anbi2d 684 . 2 (y = B → ((A C y D) ↔ (A C B D)))
61, 3, 5opkelopkabg 4245 1 ((A V B W) → (⟪A, B (C ×k D) ↔ (A C B D)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ⟪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-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-xpk 4185 This theorem is referenced by:  opkelxpk  4248  sikss1c1c  4267  opkelimagekg  4271
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