MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfsymdif3 Structured version   Visualization version   GIF version

Theorem dfsymdif3 4312
Description: Alternate definition of the symmetric difference, given in Example 4.1 of [Stoll] p. 262 (the original definition corresponds to [Stoll] p. 13). (Contributed by NM, 17-Aug-2004.) (Revised by BJ, 30-Apr-2020.)
Assertion
Ref Expression
dfsymdif3 (𝐴𝐵) = ((𝐴𝐵) ∖ (𝐴𝐵))

Proof of Theorem dfsymdif3
StepHypRef Expression
1 difin 4278 . . 3 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
2 incom 4217 . . . . 5 (𝐴𝐵) = (𝐵𝐴)
32difeq2i 4133 . . . 4 (𝐵 ∖ (𝐴𝐵)) = (𝐵 ∖ (𝐵𝐴))
4 difin 4278 . . . 4 (𝐵 ∖ (𝐵𝐴)) = (𝐵𝐴)
53, 4eqtri 2763 . . 3 (𝐵 ∖ (𝐴𝐵)) = (𝐵𝐴)
61, 5uneq12i 4176 . 2 ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵))) = ((𝐴𝐵) ∪ (𝐵𝐴))
7 difundir 4297 . 2 ((𝐴𝐵) ∖ (𝐴𝐵)) = ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))
8 df-symdif 4259 . 2 (𝐴𝐵) = ((𝐴𝐵) ∪ (𝐵𝐴))
96, 7, 83eqtr4ri 2774 1 (𝐴𝐵) = ((𝐴𝐵) ∖ (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  cdif 3960  cun 3961  cin 3962  csymdif 4258
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-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-symdif 4259
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator