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Theorem orddisj 6403
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 6383 . 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 1542  wcel 2107  cin 3948  c0 4323  {csn 4629  Ord word 6364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-eprel 5581  df-fr 5632  df-we 5634  df-ord 6368
This theorem is referenced by:  orddif  6461  omsucne  7874  tfrlem10  8387  enrefnn  9047  pssnn  9168  unfi  9172  phplem2OLD  9218  isinf  9260  isinfOLD  9261  pssnnOLD  9265  dif1ennnALT  9277  ackbij1lem5  10219  ackbij1lem14  10228  ackbij1lem16  10230  unsnen  10548  pwfi2f1o  41838
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