MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfsymdif4 Structured version   Visualization version   GIF version

Theorem dfsymdif4 4243
Description: Alternate definition of the symmetric difference. (Contributed by NM, 17-Aug-2004.) (Revised by AV, 17-Aug-2022.)
Assertion
Ref Expression
dfsymdif4 (𝐴𝐵) = {𝑥 ∣ ¬ (𝑥𝐴𝑥𝐵)}
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem dfsymdif4
StepHypRef Expression
1 elsymdif 4242 . 2 (𝑥 ∈ (𝐴𝐵) ↔ ¬ (𝑥𝐴𝑥𝐵))
21eqabi 2863 1 (𝐴𝐵) = {𝑥 ∣ ¬ (𝑥𝐴𝑥𝐵)}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1533  wcel 2098  {cab 2703  csymdif 4236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-dif 3946  df-un 3948  df-symdif 4237
This theorem is referenced by:  mbfeqalem1  25525
  Copyright terms: Public domain W3C validator