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Theorem elxpk2 4197
 Description: Membership in a cross product. (Contributed by SF, 12-Jan-2015.)
Assertion
Ref Expression
elxpk2 (A (B ×k C) ↔ x B y C A = ⟪x, y⟫)
Distinct variable groups:   x,A,y   x,B,y   x,C,y

Proof of Theorem elxpk2
StepHypRef Expression
1 ancom 437 . . 3 (((x B y C) A = ⟪x, y⟫) ↔ (A = ⟪x, y (x B y C)))
212exbii 1583 . 2 (xy((x B y C) A = ⟪x, y⟫) ↔ xy(A = ⟪x, y (x B y C)))
3 r2ex 2652 . 2 (x B y C A = ⟪x, y⟫ ↔ xy((x B y C) A = ⟪x, y⟫))
4 elxpk 4196 . 2 (A (B ×k C) ↔ xy(A = ⟪x, y (x B y C)))
52, 3, 43bitr4ri 269 1 (A (B ×k C) ↔ x B y C A = ⟪x, y⟫)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  ⟪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-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-xpk 4185 This theorem is referenced by:  xpkeq1  4198  xpkeq2  4199
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