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Theorem xpdisj2 4776
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 4498 . 2 ((𝐶 × 𝐴) ∩ (𝐷 × 𝐵)) = ((𝐶𝐷) × (𝐴𝐵))
2 xpeq2 4388 . . 3 ((𝐴𝐵) = ∅ → ((𝐶𝐷) × (𝐴𝐵)) = ((𝐶𝐷) × ∅))
3 xp0 4771 . . 3 ((𝐶𝐷) × ∅) = ∅
42, 3syl6eq 2104 . 2 ((𝐴𝐵) = ∅ → ((𝐶𝐷) × (𝐴𝐵)) = ∅)
51, 4syl5eq 2100 1 ((𝐴𝐵) = ∅ → ((𝐶 × 𝐴) ∩ (𝐷 × 𝐵)) = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  cin 2944  c0 3252   × cxp 4371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-rel 4380  df-cnv 4381
This theorem is referenced by:  xpsndisj  4777
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