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Theorem orddisj 6212
 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 6192 . 2 (Ord 𝐴 → ¬ 𝐴𝐴)
2 disjsn 4607 . 2 ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴𝐴)
31, 2sylibr 237 1 (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1538   ∈ wcel 2111   ∩ cin 3859  ∅c0 4227  {csn 4525  Ord word 6173 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5037  df-opab 5099  df-eprel 5439  df-fr 5487  df-we 5489  df-ord 6177 This theorem is referenced by:  orddif  6267  omsucne  7603  tfrlem10  8039  enrefnn  8630  phplem2  8732  pssnn  8751  unfi  8754  isinf  8782  pssnnOLD  8787  dif1enOLD  8800  ackbij1lem5  9697  ackbij1lem14  9706  ackbij1lem16  9708  unsnen  10026  pwfi2f1o  40458
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