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Theorem orddisj 4456
 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 4452 . 2 (Ord 𝐴 → ¬ 𝐴𝐴)
2 disjsn 3580 . 2 ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴𝐴)
31, 2sylibr 133 1 (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1331   ∈ wcel 1480   ∩ cin 3065  ∅c0 3358  {csn 3522  Ord word 4279 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-setind 4447 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-v 2683  df-dif 3068  df-in 3072  df-nul 3359  df-sn 3528 This theorem is referenced by:  orddif  4457  phplem2  6740  ennnfonelemhf1o  11915
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