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Theorem elxpk 4196
 Description: Membership in a Kuratowski cross product. (Contributed by SF, 12-Jan-2015.)
Assertion
Ref Expression
elxpk (A (B ×k C) ↔ xy(A = ⟪x, y (x B y C)))
Distinct variable groups:   x,A,y   x,B,y   x,C,y

Proof of Theorem elxpk
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 elex 2867 . 2 (A (B ×k C) → A V)
2 opkex 4113 . . . . 5 x, y V
3 eleq1 2413 . . . . 5 (A = ⟪x, y⟫ → (A V ↔ ⟪x, y V))
42, 3mpbiri 224 . . . 4 (A = ⟪x, y⟫ → A V)
54adantr 451 . . 3 ((A = ⟪x, y (x B y C)) → A V)
65exlimivv 1635 . 2 (xy(A = ⟪x, y (x B y C)) → A V)
7 eqeq1 2359 . . . . 5 (w = A → (w = ⟪x, y⟫ ↔ A = ⟪x, y⟫))
87anbi1d 685 . . . 4 (w = A → ((w = ⟪x, y (x B y C)) ↔ (A = ⟪x, y (x B y C))))
982exbidv 1628 . . 3 (w = A → (xy(w = ⟪x, y (x B y C)) ↔ xy(A = ⟪x, y (x B y C))))
10 df-xpk 4185 . . 3 (B ×k C) = {w xy(w = ⟪x, y (x B y C))}
119, 10elab2g 2987 . 2 (A V → (A (B ×k C) ↔ xy(A = ⟪x, y (x B y C))))
121, 6, 11pm5.21nii 342 1 (A (B ×k C) ↔ xy(A = ⟪x, y (x B y C)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859  ⟪copk 4057   ×k cxpk 4174 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-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:  elxpk2  4197  elvvk  4207  xpkvexg  4285  sikexlem  4295  insklem  4304
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