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Theorem xpdisj2 6118
Description: Cartesian products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)
Assertion
Ref Expression
xpdisj2 ((𝐴𝐵) = ∅ → ((𝐶 × 𝐴) ∩ (𝐷 × 𝐵)) = ∅)

Proof of Theorem xpdisj2
StepHypRef Expression
1 xpeq2 5658 . 2 ((𝐴𝐵) = ∅ → ((𝐶𝐷) × (𝐴𝐵)) = ((𝐶𝐷) × ∅))
2 inxp 5792 . 2 ((𝐶 × 𝐴) ∩ (𝐷 × 𝐵)) = ((𝐶𝐷) × (𝐴𝐵))
3 xp0 6114 . . 3 ((𝐶𝐷) × ∅) = ∅
43eqcomi 2742 . 2 ∅ = ((𝐶𝐷) × ∅)
51, 2, 43eqtr4g 2798 1 ((𝐴𝐵) = ∅ → ((𝐶 × 𝐴) ∩ (𝐷 × 𝐵)) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  cin 3913  c0 4286   × cxp 5635
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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-rel 5644  df-cnv 5645
This theorem is referenced by:  xpsndisj  6119
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