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Theorem xpsndisj 4971
Description: Cross products with two different singletons are disjoint. (Contributed by NM, 28-Jul-2004.)
Assertion
Ref Expression
xpsndisj (𝐵𝐷 → ((𝐴 × {𝐵}) ∩ (𝐶 × {𝐷})) = ∅)

Proof of Theorem xpsndisj
StepHypRef Expression
1 disjsn2 3592 . 2 (𝐵𝐷 → ({𝐵} ∩ {𝐷}) = ∅)
2 xpdisj2 4970 . 2 (({𝐵} ∩ {𝐷}) = ∅ → ((𝐴 × {𝐵}) ∩ (𝐶 × {𝐷})) = ∅)
31, 2syl 14 1 (𝐵𝐷 → ((𝐴 × {𝐵}) ∩ (𝐶 × {𝐷})) = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1332  wne 2309  cin 3073  c0 3366  {csn 3530   × cxp 4543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4052  ax-pow 4104  ax-pr 4137
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3076  df-un 3078  df-in 3080  df-ss 3087  df-nul 3367  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-br 3936  df-opab 3996  df-xp 4551  df-rel 4552  df-cnv 4553
This theorem is referenced by:  xp01disj  6336
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