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Theorem xpdisj2 4851
Description: Cross 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 inxp 4566 . 2 ((𝐶 × 𝐴) ∩ (𝐷 × 𝐵)) = ((𝐶𝐷) × (𝐴𝐵))
2 xpeq2 4451 . . 3 ((𝐴𝐵) = ∅ → ((𝐶𝐷) × (𝐴𝐵)) = ((𝐶𝐷) × ∅))
3 xp0 4846 . . 3 ((𝐶𝐷) × ∅) = ∅
42, 3syl6eq 2136 . 2 ((𝐴𝐵) = ∅ → ((𝐶𝐷) × (𝐴𝐵)) = ∅)
51, 4syl5eq 2132 1 ((𝐴𝐵) = ∅ → ((𝐶 × 𝐴) ∩ (𝐷 × 𝐵)) = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  cin 2998  c0 3286   × cxp 4434
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-br 3844  df-opab 3898  df-xp 4442  df-rel 4443  df-cnv 4444
This theorem is referenced by:  xpsndisj  4852
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