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Theorem xp01disjl 8124
Description: Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by Jim Kingdon, 11-Jul-2023.)
Assertion
Ref Expression
xp01disjl (({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅

Proof of Theorem xp01disjl
StepHypRef Expression
1 1n0 8122 . . 3 1o ≠ ∅
21necomi 3073 . 2 ∅ ≠ 1o
3 disjsn2 4651 . 2 (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
4 xpdisj1 6021 . 2 (({∅} ∩ {1o}) = ∅ → (({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅)
52, 3, 4mp2b 10 1 (({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1536  wne 3019  cin 3938  c0 4294  {csn 4570   × cxp 5556  1oc1o 8098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-opab 5132  df-xp 5564  df-rel 5565  df-suc 6200  df-1o 8105
This theorem is referenced by:  undjudom  9596  endjudisj  9597  djuen  9598  dju1dif  9601  dju1p1e2  9602  djucomen  9606  djuassen  9607  xpdjuen  9608  mapdjuen  9609  djudom1  9611  infdju1  9618  bj-2upln1upl  34340
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