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Theorem xpun 4421
Description: The cross product of two unions. (Contributed by NM, 12-Aug-2004.)
Assertion
Ref Expression
xpun ((𝐴𝐵) × (𝐶𝐷)) = (((𝐴 × 𝐶) ∪ (𝐴 × 𝐷)) ∪ ((𝐵 × 𝐶) ∪ (𝐵 × 𝐷)))

Proof of Theorem xpun
StepHypRef Expression
1 xpundi 4416 . 2 ((𝐴𝐵) × (𝐶𝐷)) = (((𝐴𝐵) × 𝐶) ∪ ((𝐴𝐵) × 𝐷))
2 xpundir 4417 . . 3 ((𝐴𝐵) × 𝐶) = ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶))
3 xpundir 4417 . . 3 ((𝐴𝐵) × 𝐷) = ((𝐴 × 𝐷) ∪ (𝐵 × 𝐷))
42, 3uneq12i 3125 . 2 (((𝐴𝐵) × 𝐶) ∪ ((𝐴𝐵) × 𝐷)) = (((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) ∪ ((𝐴 × 𝐷) ∪ (𝐵 × 𝐷)))
5 un4 3133 . 2 (((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) ∪ ((𝐴 × 𝐷) ∪ (𝐵 × 𝐷))) = (((𝐴 × 𝐶) ∪ (𝐴 × 𝐷)) ∪ ((𝐵 × 𝐶) ∪ (𝐵 × 𝐷)))
61, 4, 53eqtri 2106 1 ((𝐴𝐵) × (𝐶𝐷)) = (((𝐴 × 𝐶) ∪ (𝐴 × 𝐷)) ∪ ((𝐵 × 𝐶) ∪ (𝐵 × 𝐷)))
Colors of variables: wff set class
Syntax hints:   = wceq 1285  cun 2972   × cxp 4363
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-opab 3842  df-xp 4371
This theorem is referenced by: (None)
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