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Theorem orddisj 5905
 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 5884 . 2 (Ord 𝐴 → ¬ 𝐴𝐴)
2 disjsn 4381 . 2 ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴𝐴)
31, 2sylibr 224 1 (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1630   ∈ wcel 2144   ∩ cin 3720  ∅c0 4061  {csn 4314  Ord word 5865 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-eprel 5162  df-fr 5208  df-we 5210  df-ord 5869 This theorem is referenced by:  orddif  5963  tfrlem10  7635  phplem2  8295  isinf  8328  pssnn  8333  dif1en  8348  ackbij1lem5  9247  ackbij1lem14  9256  ackbij1lem16  9258  unsnen  9576  pwfi2f1o  38185
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