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Theorem xpdisj1 6170
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 5696 . 2 ((𝐴𝐵) = ∅ → ((𝐴𝐵) × (𝐶𝐷)) = (∅ × (𝐶𝐷)))
2 inxp 5838 . 2 ((𝐴 × 𝐶) ∩ (𝐵 × 𝐷)) = ((𝐴𝐵) × (𝐶𝐷))
3 0xp 5780 . . 3 (∅ × (𝐶𝐷)) = ∅
43eqcomi 2737 . 2 ∅ = (∅ × (𝐶𝐷))
51, 2, 43eqtr4g 2793 1 ((𝐴𝐵) = ∅ → ((𝐴 × 𝐶) ∩ (𝐵 × 𝐷)) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  cin 3948  c0 4326   × cxp 5680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-opab 5215  df-xp 5688  df-rel 5689
This theorem is referenced by:  djudisj  6176  xp01disjl  8519  djuin  9949  nosupbnd2lem1  27668  noetasuplem3  27688  noetasuplem4  27689  xpdisjres  32409  esum2dlem  33744  disjxp1  44464
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