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Theorem xpun 5370
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 5365 . 2 ((𝐴𝐵) × (𝐶𝐷)) = (((𝐴𝐵) × 𝐶) ∪ ((𝐴𝐵) × 𝐷))
2 xpundir 5366 . . 3 ((𝐴𝐵) × 𝐶) = ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶))
3 xpundir 5366 . . 3 ((𝐴𝐵) × 𝐷) = ((𝐴 × 𝐷) ∪ (𝐵 × 𝐷))
42, 3uneq12i 3958 . 2 (((𝐴𝐵) × 𝐶) ∪ ((𝐴𝐵) × 𝐷)) = (((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) ∪ ((𝐴 × 𝐷) ∪ (𝐵 × 𝐷)))
5 un4 3966 . 2 (((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) ∪ ((𝐴 × 𝐷) ∪ (𝐵 × 𝐷))) = (((𝐴 × 𝐶) ∪ (𝐴 × 𝐷)) ∪ ((𝐵 × 𝐶) ∪ (𝐵 × 𝐷)))
61, 4, 53eqtri 2828 1 ((𝐴𝐵) × (𝐶𝐷)) = (((𝐴 × 𝐶) ∪ (𝐴 × 𝐷)) ∪ ((𝐵 × 𝐶) ∪ (𝐵 × 𝐷)))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1637  cun 3761   × cxp 5303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-v 3389  df-un 3768  df-opab 4900  df-xp 5311
This theorem is referenced by:  ex-xp  27618
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