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Theorem xpsndisj 5459
Description: Cartesian 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 4189 . 2 (𝐵𝐷 → ({𝐵} ∩ {𝐷}) = ∅)
2 xpdisj2 5458 . 2 (({𝐵} ∩ {𝐷}) = ∅ → ((𝐴 × {𝐵}) ∩ (𝐶 × {𝐷})) = ∅)
31, 2syl 17 1 (𝐵𝐷 → ((𝐴 × {𝐵}) ∩ (𝐶 × {𝐷})) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wne 2776  cin 3535  c0 3870  {csn 4121   × cxp 5023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pr 4825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-br 4575  df-opab 4635  df-xp 5031  df-rel 5032  df-cnv 5033
This theorem is referenced by:  xp01disj  7437  unxpdom2  8027  sucxpdom  8028
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