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Theorem xpun 5762
Description: The Cartesian 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 5757 . 2 ((𝐴𝐵) × (𝐶𝐷)) = (((𝐴𝐵) × 𝐶) ∪ ((𝐴𝐵) × 𝐷))
2 xpundir 5758 . . 3 ((𝐴𝐵) × 𝐶) = ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶))
3 xpundir 5758 . . 3 ((𝐴𝐵) × 𝐷) = ((𝐴 × 𝐷) ∪ (𝐵 × 𝐷))
42, 3uneq12i 4176 . 2 (((𝐴𝐵) × 𝐶) ∪ ((𝐴𝐵) × 𝐷)) = (((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) ∪ ((𝐴 × 𝐷) ∪ (𝐵 × 𝐷)))
5 un4 4185 . 2 (((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) ∪ ((𝐴 × 𝐷) ∪ (𝐵 × 𝐷))) = (((𝐴 × 𝐶) ∪ (𝐴 × 𝐷)) ∪ ((𝐵 × 𝐶) ∪ (𝐵 × 𝐷)))
61, 4, 53eqtri 2767 1 ((𝐴𝐵) × (𝐶𝐷)) = (((𝐴 × 𝐶) ∪ (𝐴 × 𝐷)) ∪ ((𝐵 × 𝐶) ∪ (𝐵 × 𝐷)))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  cun 3961   × cxp 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-opab 5211  df-xp 5695
This theorem is referenced by:  ex-xp  30465
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