MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  disjcsn Structured version   Visualization version   GIF version

Theorem disjcsn 9619
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 9612 . 2 ¬ 𝐴𝐴
2 disjsn 4711 . 2 ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴𝐴)
31, 2mpbir 230 1 (𝐴 ∩ {𝐴}) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1534  wcel 2099  cin 3943  c0 4318  {csn 4624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-pr 5423  ax-reg 9607
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-nul 4319  df-sn 4625  df-pr 4627
This theorem is referenced by:  bnj927  34336  bnj535  34457  sucdifsn2  37643  ressucdifsn2  37649
  Copyright terms: Public domain W3C validator