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Theorem orddisj 6401
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 6381 . 2 (Ord 𝐴 → ¬ 𝐴𝐴)
2 disjsn 4714 . 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 1539  wcel 2104  cin 3946  c0 4321  {csn 4627  Ord word 6362
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-eprel 5579  df-fr 5630  df-we 5632  df-ord 6366
This theorem is referenced by:  orddif  6459  omsucne  7876  tfrlem10  8389  enrefnn  9049  pssnn  9170  unfi  9174  phplem2OLD  9220  isinf  9262  isinfOLD  9263  pssnnOLD  9267  dif1ennnALT  9279  ackbij1lem5  10221  ackbij1lem14  10230  ackbij1lem16  10232  unsnen  10550  pwfi2f1o  42140
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