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Theorem xpun 5710
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 5705 . 2 ((𝐴𝐵) × (𝐶𝐷)) = (((𝐴𝐵) × 𝐶) ∪ ((𝐴𝐵) × 𝐷))
2 xpundir 5706 . . 3 ((𝐴𝐵) × 𝐶) = ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶))
3 xpundir 5706 . . 3 ((𝐴𝐵) × 𝐷) = ((𝐴 × 𝐷) ∪ (𝐵 × 𝐷))
42, 3uneq12i 4126 . 2 (((𝐴𝐵) × 𝐶) ∪ ((𝐴𝐵) × 𝐷)) = (((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) ∪ ((𝐴 × 𝐷) ∪ (𝐵 × 𝐷)))
5 un4 4134 . 2 (((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) ∪ ((𝐴 × 𝐷) ∪ (𝐵 × 𝐷))) = (((𝐴 × 𝐶) ∪ (𝐴 × 𝐷)) ∪ ((𝐵 × 𝐶) ∪ (𝐵 × 𝐷)))
61, 4, 53eqtri 2769 1 ((𝐴𝐵) × (𝐶𝐷)) = (((𝐴 × 𝐶) ∪ (𝐴 × 𝐷)) ∪ ((𝐵 × 𝐶) ∪ (𝐵 × 𝐷)))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  cun 3913   × cxp 5636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3450  df-un 3920  df-opab 5173  df-xp 5644
This theorem is referenced by:  ex-xp  29422
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