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Theorem disjcsn 9642
Description: A class is disjoint from its singleton. A consequence of regularity. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by BJ, 4-Apr-2019.)
Assertion
Ref Expression
disjcsn (𝐴 ∩ {𝐴}) = ∅

Proof of Theorem disjcsn
StepHypRef Expression
1 elirr 9635 . 2 ¬ 𝐴𝐴
2 disjsn 4716 . 2 ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴𝐴)
31, 2mpbir 231 1 (𝐴 ∩ {𝐴}) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1537  wcel 2106  cin 3962  c0 4339  {csn 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-pr 5438  ax-reg 9630
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-nul 4340  df-sn 4632  df-pr 4634
This theorem is referenced by:  bnj927  34762  bnj535  34883  sucdifsn2  38219  ressucdifsn2  38225
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