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Theorem orddisj 6407
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 6387 . 2 (Ord 𝐴 → ¬ 𝐴𝐴)
2 disjsn 4716 . 2 ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴𝐴)
31, 2sylibr 233 1 (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1534  wcel 2099  cin 3946  c0 4323  {csn 4629  Ord word 6368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-eprel 5582  df-fr 5633  df-we 5635  df-ord 6372
This theorem is referenced by:  orddif  6465  omsucne  7889  tfrlem10  8408  enrefnn  9072  pssnn  9193  unfi  9197  phplem2OLD  9243  isinf  9285  isinfOLD  9286  pssnnOLD  9290  dif1ennnALT  9302  ackbij1lem5  10248  ackbij1lem14  10257  ackbij1lem16  10259  unsnen  10577  pwfi2f1o  42520
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