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Theorem orddisj 6424
Description: An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.)
Assertion
Ref Expression
orddisj (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅)

Proof of Theorem orddisj
StepHypRef Expression
1 ordirr 6404 . 2 (Ord 𝐴 → ¬ 𝐴𝐴)
2 disjsn 4716 . 2 ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴𝐴)
31, 2sylibr 234 1 (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1537  wcel 2106  cin 3962  c0 4339  {csn 4631  Ord word 6385
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-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-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-eprel 5589  df-fr 5641  df-we 5643  df-ord 6389
This theorem is referenced by:  orddif  6482  omsucne  7906  tfrlem10  8426  enrefnn  9086  pssnn  9207  unfi  9210  phplem2OLD  9253  isinf  9294  isinfOLD  9295  dif1ennnALT  9309  ackbij1lem5  10261  ackbij1lem14  10270  ackbij1lem16  10272  unsnen  10591  pwfi2f1o  43085
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