MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  symdif2 Structured version   Unicode version

Theorem symdif2 3701
Description: Two ways to express symmetric difference. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
symdif2  |-  ( ( A  \  B )  u.  ( B  \  A ) )  =  { x  |  -.  ( x  e.  A  <->  x  e.  B ) }
Distinct variable groups:    x, A    x, B

Proof of Theorem symdif2
StepHypRef Expression
1 eldif 3446 . . . 4  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
2 eldif 3446 . . . 4  |-  ( x  e.  ( B  \  A )  <->  ( x  e.  B  /\  -.  x  e.  A ) )
31, 2orbi12i 523 . . 3  |-  ( ( x  e.  ( A 
\  B )  \/  x  e.  ( B 
\  A ) )  <-> 
( ( x  e.  A  /\  -.  x  e.  B )  \/  (
x  e.  B  /\  -.  x  e.  A
) ) )
4 elun 3606 . . 3  |-  ( x  e.  ( ( A 
\  B )  u.  ( B  \  A
) )  <->  ( x  e.  ( A  \  B
)  \/  x  e.  ( B  \  A
) ) )
5 xor 899 . . 3  |-  ( -.  ( x  e.  A  <->  x  e.  B )  <->  ( (
x  e.  A  /\  -.  x  e.  B
)  \/  ( x  e.  B  /\  -.  x  e.  A )
) )
63, 4, 53bitr4i 280 . 2  |-  ( x  e.  ( ( A 
\  B )  u.  ( B  \  A
) )  <->  -.  (
x  e.  A  <->  x  e.  B ) )
76abbi2i 2550 1  |-  ( ( A  \  B )  u.  ( B  \  A ) )  =  { x  |  -.  ( x  e.  A  <->  x  e.  B ) }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872   {cab 2407    \ cdif 3433    u. cun 3434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-v 3082  df-dif 3439  df-un 3441
This theorem is referenced by:  mbfeqalem  22596
  Copyright terms: Public domain W3C validator