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Theorem xpdisj1 4747
 Description: Cross 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 inxp 4470 . 2 ((𝐴 × 𝐶) ∩ (𝐵 × 𝐷)) = ((𝐴𝐵) × (𝐶𝐷))
2 xpeq1 4359 . . 3 ((𝐴𝐵) = ∅ → ((𝐴𝐵) × (𝐶𝐷)) = (∅ × (𝐶𝐷)))
3 0xp 4420 . . 3 (∅ × (𝐶𝐷)) = ∅
42, 3syl6eq 2088 . 2 ((𝐴𝐵) = ∅ → ((𝐴𝐵) × (𝐶𝐷)) = ∅)
51, 4syl5eq 2084 1 ((𝐴𝐵) = ∅ → ((𝐴 × 𝐶) ∩ (𝐵 × 𝐷)) = ∅)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ∩ cin 2916  ∅c0 3224   × cxp 4343 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-xp 4351  df-rel 4352 This theorem is referenced by:  djudisj  4750
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