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Theorem disjcsn 9644
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 9637 . 2 ¬ 𝐴𝐴
2 disjsn 4711 . 2 ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴𝐴)
31, 2mpbir 231 1 (𝐴 ∩ {𝐴}) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wcel 2108  cin 3950  c0 4333  {csn 4626
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-pr 5432  ax-reg 9632
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-nul 4334  df-sn 4627  df-pr 4629
This theorem is referenced by:  bnj927  34783  bnj535  34904  sucdifsn2  38239  ressucdifsn2  38245
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