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

Theorem symdifcom 4246
Description: Symmetric difference commutes. (Contributed by Scott Fenton, 24-Apr-2012.)
Assertion
Ref Expression
symdifcom (𝐴𝐵) = (𝐵𝐴)

Proof of Theorem symdifcom
StepHypRef Expression
1 uncom 4154 . 2 ((𝐴𝐵) ∪ (𝐵𝐴)) = ((𝐵𝐴) ∪ (𝐴𝐵))
2 df-symdif 4245 . 2 (𝐴𝐵) = ((𝐴𝐵) ∪ (𝐵𝐴))
3 df-symdif 4245 . 2 (𝐵𝐴) = ((𝐵𝐴) ∪ (𝐴𝐵))
41, 2, 33eqtr4i 2766 1 (𝐴𝐵) = (𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  cdif 3946  cun 3947  csymdif 4244
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 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3475  df-un 3954  df-symdif 4245
This theorem is referenced by:  symdifeq2  4248
  Copyright terms: Public domain W3C validator