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Theorem xp01disj 8528
Description: Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by NM, 2-Jun-2007.)
Assertion
Ref Expression
xp01disj ((𝐴 × {∅}) ∩ (𝐶 × {1o})) = ∅

Proof of Theorem xp01disj
StepHypRef Expression
1 1n0 8525 . . 3 1o ≠ ∅
21necomi 2993 . 2 ∅ ≠ 1o
3 xpsndisj 6185 . 2 (∅ ≠ 1o → ((𝐴 × {∅}) ∩ (𝐶 × {1o})) = ∅)
42, 3ax-mp 5 1 ((𝐴 × {∅}) ∩ (𝐶 × {1o})) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wne 2938  cin 3962  c0 4339  {csn 4631   × cxp 5687  1oc1o 8498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-suc 6392  df-1o 8505
This theorem is referenced by:  endisj  9097
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