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Theorem xpdisj1 6181
Description: Cartesian products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)
Assertion
Ref Expression
xpdisj1 ((𝐴𝐵) = ∅ → ((𝐴 × 𝐶) ∩ (𝐵 × 𝐷)) = ∅)

Proof of Theorem xpdisj1
StepHypRef Expression
1 xpeq1 5699 . 2 ((𝐴𝐵) = ∅ → ((𝐴𝐵) × (𝐶𝐷)) = (∅ × (𝐶𝐷)))
2 inxp 5842 . 2 ((𝐴 × 𝐶) ∩ (𝐵 × 𝐷)) = ((𝐴𝐵) × (𝐶𝐷))
3 0xp 5784 . . 3 (∅ × (𝐶𝐷)) = ∅
43eqcomi 2746 . 2 ∅ = (∅ × (𝐶𝐷))
51, 2, 43eqtr4g 2802 1 ((𝐴𝐵) = ∅ → ((𝐴 × 𝐶) ∩ (𝐵 × 𝐷)) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  cin 3950  c0 4333   × cxp 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-opab 5206  df-xp 5691  df-rel 5692
This theorem is referenced by:  djudisj  6187  xp01disjl  8530  djuin  9958  nosupbnd2lem1  27760  noetasuplem3  27780  noetasuplem4  27781  xpdisjres  32611  esum2dlem  34093  disjxp1  45074
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